- Identify the total number of trials from the binomial coefficient. - The lower number in the binomial coefficient is the success count. - The base raised to the success-count exponent is \(p\). - The two exponents add to \(n\).

1. Compare each expression with \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\). 2. In part a), \(n=20\), \(p=0.75\), and \(k=11\). 3. In part b), \(n=50\), \(p=0.05\), and \(k=2\). 4. In part c), \(n=9\), \(p=\frac14\), and \(k=5\).

a) \(n=20\), \(p=0.75\), and \(k=11\) b) \(n=50\), \(p=0.05\), and \(k=2\) c) \(n=9\), \(p=\frac14\), and \(k=5\)

- Match the values with the standard binomial probability formula. - Find the failure probability from \(1-p\). - Subtract the number of successes from the number of trials. - The success and failure exponents sum to \(n\).

1. The binomial probability formula is \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\). 2. Substituting the values gives \(P(X=4)=\binom{16}{4}\left(\frac13\right)^4\left(\frac23\right)^{12}\). 3. The exponent on the failure probability is \(n-k=16-4=12\), the number of failures.

a) \(P(X=4)=\binom{16}{4}\left(\frac13\right)^4\left(\frac23\right)^{12}\) b) The exponent is \(12\), found from \(n-k\).

- Check whether the number of trials is fixed in advance. - Ask whether each trial has two relevant outcomes with the same success probability. - Check whether the trial outcomes can reasonably be treated as independent.

1. Situation A has a fixed number of trials, two relevant outcomes on each trial, independent trials, and a constant success probability. 2. Therefore, \(X\sim\operatorname{Bin}(10,0.65).\) 3. Situation B is a waiting-time setting, situation C changes probabilities because sampling is without replacement, and situation D does not describe a fixed number of Bernoulli trials.

Situation A. The parameters are \(n=10\) and \(p=0.65\).

- What counts as success on one trial? - Check whether the number of trials is fixed. - Consider whether one result changes the chance of success on the next attempt.

1. There is a fixed number of trials: \(10\) shots. 2. Each shot is either made or missed, so there are two possible outcomes. 3. The shots are independent, and the success probability stays \(0.6\). 4. Thus, \(X\) is binomial with \(n=10\) and \(p=0.6\).

\(X\) is binomial because it counts successes in \(10\) independent two-outcome trials, each with success probability \(0.6\). So \(n=10\), \(p=0.6\).

- Recall which parameter names appear inside the binomial notation. - A success count cannot exceed the number of trials. - Consider whether a count can be a fraction.

1. In the notation \(\operatorname{Bin}(n,p)\), \(n\) is the fixed number of trials and \(p\) is the success probability on each trial. 2. Here \(n=12\) and \(p=0.3\). 3. \(X\) counts successes, so it can take each integer value from \(0\) through \(12\).

\(12\) is the number of trials, and \(0.3\) is the probability of success on each trial. Possible values are \(0,1,2,\dots,12\).

- Check each condition of a binomial model, not just independence. - Compare the success chance on the first and second shot. - Does a binomial model allow that chance to change between trials?

1. Each trial has two outcomes, and there are two independent shots. 2. The chance of success differs between the shots, so the trials do not have a common success probability. 3. The count is not binomial.

No. The success probabilities are different on the two shots.

- Match each part of the expression with the standard binomial probability formula. - The success and failure exponents add to the total number of trials. - The two outcome probabilities add to \(1\).

1. Comparing the expression with \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\), the number of successes is \(k=6\). 2. The failure exponent is \(m=n-k=15-6=9\). 3. Since \(1-p=0.75\), \(p=0.25\). 4. Therefore, \(P(X=6)=\binom{15}{6}(0.25)^6(0.75)^9\approx0.0917\).

\(k=6\), \(p=0.25\), \(m=9\), and \(P(X=6)\approx0.0917\)

- The two exponents in a binomial probability add to \(n\). - The lower number in \(\binom{n}{k}\) is the number of successes. - The success and failure probabilities add to \(1\).

1. In a), the success count is \(3\), so the first two blanks are \(3\). The failure exponent is \(10-3=7\), so the third blank is \(7\). 2. In b), the lower number in the binomial coefficient is the success count, so the first blank is \(5\). The total number of trials is \(5+7=12\), and the failure probability is \(1-\frac{1}{3}=\frac{2}{3}\), so the second blank is \(\frac{2}{3}\). 3. In c), the success and failure exponents add to \(n\). Thus, \(n=2+8=10\), so the blank is \(10\).

a) From left to right: \(3\), \(3\), \(7\). b) From left to right: \(5\), \(\frac{2}{3}\); also \(n=12\). c) \(10\).

- Find the probability of a failure on one trial. - Use the binomial coefficient to count the placements of the four successes. - Apply the exponents carefully. - Round only the final result.

1. Here, \(n=12\), \(k=4\), \(p=\frac14\), and \(1-p=\frac34\). 2. Use the binomial probability formula: \(P(X=4)=\binom{12}{4}\left(\frac14\right)^4\left(\frac34\right)^8\). 3. Since \(\binom{12}{4}=495\), the probability is \(495\cdot(0.25)^4\cdot(0.75)^8\approx0.1935777\). 4. Rounded to four decimal places, \(P(X=4)\approx0.1936\).

\(P(X=4)\approx0.1936\)

- Identify \(n\) and \(p\). - Use the binomial probability formula for each fixed number of hits. - Scientific notation is useful for the very small zero-hit probability.

1. Let \(X\sim\operatorname{Bin}(12,0.80)\). 2. \(P(X=10)=\binom{12}{10}(0.8)^{10}(0.2)^2\approx0.2835.\) 3. \(P(X=8)=\binom{12}{8}(0.8)^8(0.2)^4\approx0.1329.\) 4. \(P(X=0)=(0.2)^{12}=4.096\times10^{-9}.\)

a) \(\binom{12}{10}(0.8)^{10}(0.2)^2\approx0.2835\). b) \(\binom{12}{8}(0.8)^8(0.2)^4\approx0.1329\). c) \(4.096\times10^{-9}\).

- Exactly four nondefective components means exactly one defective component. - Use the binomial probability formula for each machine separately. - Compare the two probabilities after calculating them.

1. For Machine A, \(P(X=4)=\binom54(0.9)^4(0.1)=0.32805\). 2. For Machine B, \(P(X=4)=\binom54(0.7)^4(0.3)=0.36015\). 3. Since \(0.36015>0.32805\), exactly four nondefective components are more likely for Machine B.

Machine B. Its probability is \(0.36015\), compared with \(0.32805\) for Machine A.

- Identify the number of trials and success probability for each archer. - Count the arrangements that give Jordan exactly two hits. - There is only one outcome pattern in which Leah hits all three times. - Compare the resulting probabilities.

1. For Jordan, \(P(X=2)=\binom32(0.5)^2(0.5)=0.375\). 2. For Leah, \(P(Y=3)=(0.75)^3=0.421875\). 3. Since \(0.421875>0.375\), Leah getting three center hits is more likely.

Leah hitting the center on all \(3\) shots is more likely: \(0.421875\) compared with \(0.375\).

- Identify \(n\), \(p\), and \(k\). - Find the probability of a missed free throw. - Use the binomial formula for an exact number of successes. - Include the binomial coefficient for the arrangements of the three made shots.

1. The parameters are \(n=5\), \(p=0.6\), and \(k=3\), with failure probability \(0.4\). 2. \(P(X=3)=\binom53(0.6)^3(0.4)^2\). 3. Since \(\binom53=10\), \((0.6)^3=0.216\), and \((0.4)^2=0.16\), the probability is \(10\cdot0.216\cdot0.16=0.3456\).

\(P(X=3)=0.3456\), or \(34.56\%\)

- Define a success as selecting a defective component. - Identify \(n\), \(p\), and \(k\). - Use the binomial formula for exactly \(2\) successes.

1. Let \(X\sim\operatorname{Bin}(10,0.2)\). 2. \(P(X=2)=\binom{10}{2}(0.2)^2(0.8)^8.\) 3. Since \(\binom{10}{2}=45\), \(P(X=2)=0.301989888.\)

\(P(X=2)=0.301989888\).

- Model the count with a binomial distribution. - “At most one” includes \(0\) and \(1\). - Keep exact expressions until the final rounding.

1. Let \(X\sim\operatorname{Bin}(30,0.05)\). 2. \(P(X=0)=0.95^{30}\approx0.2146.\) 3. \(P(X\le1)=0.95^{30}+30(0.05)(0.95)^{29}\approx0.5535.\)

a) \(0.95^{30}\approx0.2146\). b) \(0.95^{30}+30(0.05)(0.95)^{29}\approx0.5535\).

- Identify \(n\) and \(p\). - “At least \(4\)” includes both \(4\) and \(5\) hits. - Keep exact expressions until the final rounding.

1. Let \(X\sim\operatorname{Bin}(5,0.85)\). 2. \(P(X=4)=\binom54(0.85)^4(0.15)\approx0.3915.\) 3. \(P(X\ge4)=\binom54(0.85)^4(0.15)+(0.85)^5\approx0.8352.\)

a) \(\binom54(0.85)^4(0.15)\approx0.3915\). b) \(\binom54(0.85)^4(0.15)+(0.85)^5\approx0.8352\).

- Check that the hit probability stays constant from shot to shot. - Identify \(n\), \(p\), and \(k\). - Use the binomial probability formula. - Include the number of arrangements of the two hits.

1. The parameters are \(n=4\), \(p=0.6\), and \(k=2\). 2. The number of arrangements of two hits among four shots is \(\binom42=6\). 3. The probability is \(P(X=2)=\binom42(0.6)^2(0.4)^2=6\cdot0.36\cdot0.16=0.3456\).

\(P(X=2)=0.3456\), or \(34.56\%\)

- Model each seed as a success-or-failure trial. - Identify the binomial parameters. - Count the arrangements that contain exactly five germinations. - Include the probability of the two seeds that do not germinate.

1. The parameters are \(n=7\), \(p=0.8\), and \(k=5\). 2. The number of arrangements is \(\binom75=21\). 3. The probability of one arrangement with five germinations and two failures is \((0.8)^5(0.2)^2=0.0131072\). 4. Therefore, \(P(X=5)=21\cdot0.0131072=0.2752512\).

\(P(X=5)=0.2752512\), or about \(27.53\%\)

- Identify \(n\), \(p\), and \(k\) for each part. - Find the failure probability \(1-p\). - Evaluate the binomial coefficient before multiplying.

1. For part a), \(P(X=3)=\binom53(0.4)^3(0.6)^2=10\cdot0.064\cdot0.36=0.2304\). 2. For part b), \(P(X=2)=\binom62(0.1)^2(0.9)^4=15\cdot0.01\cdot0.6561=0.098415\approx0.0984\). 3. For part c), \(P(X=0)=\binom40(0.5)^0(0.5)^4=0.0625\).

a) \(0.2304\) b) \(0.0984\) c) \(0.0625\)

- Calculate both probabilities separately. - Keep enough decimal places to compare them reliably. - Compare the final values directly.

1. \(P(X=1)=\binom{10}{1}(0.2)(0.8)^9=0.268435456\). 2. \(P(X=2)=\binom{10}{2}(0.2)^2(0.8)^8=0.301989888\). 3. Since \(0.301989888>0.268435456\), \(X=2\) is more likely.

\(X=2\) is more likely: about \(30.2\%\), compared with about \(26.8\%\) for \(X=1\).

- What do the exponents tell you about the total number of trials? - What does \(\binom{12}{2}\) tell you about the number of successes? - What familiar random device gives a success probability of \(\frac{1}{6}\)? - Decide which outcome will count as a success.

1. The expression has the binomial form \(\binom{n}{k}p^k(1-p)^{n-k}\), with \(n=12\), \(k=2\), and \(p=\frac{1}{6}\). 2. A suitable experiment is rolling a fair six-sided die \(12\) times and defining a success as rolling a \(6\). 3. Therefore, event \(A\) is rolling exactly two \(6\)s in the \(12\) rolls.

Random experiment: Roll a fair six-sided die \(12\) times. Event \(A\): Exactly two of the rolls show a \(6\).

- Identify \(n\), \(p\), and \(k\) for each probability. - Many calculators use a command such as binompdf for an exact binomial probability. - Use the fifth decimal place to round to four decimal places.

1. For part a), \(P(X=3)=\binom{15}{3}(0.2)^3(0.8)^{12}\approx0.2501\). 2. For part b), \(P(X=10)=\binom{20}{10}(0.45)^{10}(0.55)^{10}\approx0.1593\). 3. For part c), \(P(X=2)=\binom{50}{2}(0.05)^2(0.95)^{48}\approx0.2611\).

a) \(0.2501\) b) \(0.1593\) c) \(0.2611\)

- Find the probability of rolling a \(6\) on one roll. - Identify the total number of trials. - Match each event to the required number of successes. - Use the binomial probability formula or your calculator's exact binomial probability command.

1. Let \(X\) be the number of rolls that show a \(6\). Then \(X\) is binomial with \(n=10\) and \(p=\frac{1}{6}\). 2. For part a), \(P(X=2)=\binom{10}{2}\left(\frac{1}{6}\right)^2\left(\frac{5}{6}\right)^8\approx0.2907\). 3. For part b), \(P(X=5)=\binom{10}{5}\left(\frac{1}{6}\right)^5\left(\frac{5}{6}\right)^5\approx0.0130\). 4. For part c), \(P(X=0)=\binom{10}{0}\left(\frac{1}{6}\right)^0\left(\frac{5}{6}\right)^{10}\approx0.1615\).

a) \(0.2907\) b) \(0.0130\) c) \(0.1615\)

- Find the probability of one specified face on a fair four-sided die. - Identify \(n\), \(k\), and \(p\). - Include the probability of the six rolls that are not \(3\)s.

1. Since the four outcomes are equally likely, \(p=\frac{1}{4}=0.25\), and \(1-p=0.75\). 2. \(P(X=2)=\binom{8}{2}(0.25)^2(0.75)^6\). 3. Since \(\binom{8}{2}=28\), \(P(X=2)=0.31146240234375\).

a) \(p=0.25\). b) \(P(X=2)=\binom{8}{2}(0.25)^2(0.75)^6=0.31146240234375\).

- Identify the total number of trials, the number of successes, and the success probability. - Decide whether the question asks for one exact value or a cumulative probability. - Use your calculator's exact binomial probability command.

1. The binomial parameters are \(n=100\), \(p=0.65\), and \(k=60\). 2. The required exact probability is \(P(X=60)=\binom{100}{60}(0.65)^{60}(0.35)^{40}\). 3. A calculator gives \(P(X=60)\approx0.0474\).

\(P(X=60)\approx0.0474\)

- What operation corresponds to the word “or” when the outcomes cannot happen at the same time? - Which values of \(k\) are included in “at least \(4\)”? - How are the events “at least \(4\)” and “at most \(3\)” related? - How can you use a complement when the total probability is \(1\)?

1. For Event A, add the probabilities for \(k=5\) and \(k=6\): \(P(A)=0.3932+0.2621=0.6553\). 2. For Event B, add the probabilities for \(k=4\), \(k=5\), and \(k=6\): \(P(B)=0.2458+0.3932+0.2621=0.9011\). 3. Event C is the complement of Event B, so \(P(C)=1-P(B)=1-0.9011=0.0989\).

Event A: \(P(A)=0.6553\) Event B: \(P(B)=0.9011\) Event C: \(P(C)=0.0989\)

- What values can \(X\) take in three trials? - Use the formula for the probability of exactly \(k\) successes in \(n\) independent trials. - Find the probability of a failure before substituting values. - A probability distribution table pairs each possible value of \(X\) with its probability.

1. The success probability is \(p=0.4\), the failure probability is \(1-p=0.6\), and the number of trials is \(n=3\). 2. Use the binomial probability formula \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\) for \(k\in\{0,1,2,3\}\). 3. \(P(X=0)=\binom{3}{0}(0.4)^0(0.6)^3=0.216\). 4. \(P(X=1)=\binom{3}{1}(0.4)^1(0.6)^2=0.432\). 5. \(P(X=2)=\binom{3}{2}(0.4)^2(0.6)^1=0.288\). 6. \(P(X=3)=\binom{3}{3}(0.4)^3(0.6)^0=0.064\).

<table> <tr> <td>\(k\)</td> <td>\(0\)</td> <td>\(1\)</td> <td>\(2\)</td> <td>\(3\)</td> </tr> <tr> <td>\(P(X=k)\)</td> <td>\(0.216\)</td> <td>\(0.432\)</td> <td>\(0.288\)</td> <td>\(0.064\)</td> </tr> </table>

- Use the binomial probability formula for each value of \(k\). - The binomial coefficient \(\binom{n}{k}\) counts the arrangements with exactly \(k\) successes. - Recall what the probabilities in any complete distribution must add to. - Track the exponent on the failure probability carefully.

1. The parameters are \(n=5\), \(p=0.2\), and \(1-p=0.8\). 2. Apply \(P(X=k)=\binom{5}{k}(0.2)^k(0.8)^{5-k}\): \(P(X=0)=\binom{5}{0}(0.2)^0(0.8)^5=0.32768\) \(P(X=1)=\binom{5}{1}(0.2)^1(0.8)^4=0.4096\) \(P(X=2)=\binom{5}{2}(0.2)^2(0.8)^3=0.2048\) \(P(X=3)=\binom{5}{3}(0.2)^3(0.8)^2=0.0512\) \(P(X=4)=\binom{5}{4}(0.2)^4(0.8)^1=0.0064\) \(P(X=5)=\binom{5}{5}(0.2)^5(0.8)^0=0.00032\) 3. The sum is \(0.32768+0.4096+0.2048+0.0512+0.0064+0.00032=1\), as required for a probability distribution.

a) \(P(X=0)=0.32768\); \(P(X=1)=0.4096\); \(P(X=2)=0.2048\); \(P(X=3)=0.0512\); \(P(X=4)=0.0064\); \(P(X=5)=0.00032\). b) \(0.32768+0.4096+0.2048+0.0512+0.0064+0.00032=1\).

- Identify the probability of drawing a red ball on one draw. - Because the draws are with replacement, use the same success probability for both trials. - Use the binomial probability formula for \(X=0\), \(X=1\), and \(X=2\). - For part b), consider using the complement of drawing at least one red ball.

1. On each draw, \(P(\text{red})=\frac{5}{8}\) and \(P(\text{yellow})=\frac{3}{8}\). Because the ball is replaced, the draws are independent. 2. Use the binomial probability formula with \(n=2\) and \(p=\frac{5}{8}\): \(P(X=0)=\binom{2}{0}\left(\frac{5}{8}\right)^0\left(\frac{3}{8}\right)^2=\frac{9}{64}\). \(P(X=1)=\binom{2}{1}\left(\frac{5}{8}\right)^1\left(\frac{3}{8}\right)^1=\frac{30}{64}=\frac{15}{32}\). \(P(X=2)=\binom{2}{2}\left(\frac{5}{8}\right)^2\left(\frac{3}{8}\right)^0=\frac{25}{64}\). 3. For at least one red ball, use the complement: \(P(X\ge1)=1-P(X=0)=1-\frac{9}{64}=\frac{55}{64}\).

a) <table border="1"> <tr> <td>\(x\)</td> <td>\(0\)</td> <td>\(1\)</td> <td>\(2\)</td> </tr> <tr> <td>\(P(X=x)\)</td> <td>\(\frac{9}{64}\)</td> <td>\(\frac{30}{64}\)</td> <td>\(\frac{25}{64}\)</td> </tr> </table> b) \(P(X\ge1)=\frac{55}{64}\approx85.9\%\)

- Find the probability of red on one spin. - Distinguish between one specified order and any order with the same number of red outcomes. - Use a binomial coefficient to count arrangements when the positions are not specified. - For no red outcomes, multiply the probability of not red across all spins.

1. The probability of red is \(p=0.25\), and the probability of not red is \(0.75\). 2. For part a), the specified sequence is red, red, not red, not red, not red, not red. Its probability is \((0.25)^2(0.75)^4=\frac{81}{4096}\approx0.0198\). 3. For part b), the two red outcomes can occur in \(\binom62=15\) arrangements. Therefore, \(P(X=2)=\binom62(0.25)^2(0.75)^4=\frac{1215}{4096}\approx0.2966\). 4. For part c), \(P(X=0)=(0.75)^6=\frac{729}{4096}\approx0.1780\).

a) \(\frac{81}{4096}\approx0.0198\) b) \(\frac{1215}{4096}\approx0.2966\) c) \(\frac{729}{4096}\approx0.1780\)

- For one randomly guessed question, what is the chance of choosing the correct answer? - How many trials are there, and how many of them must be successes? - How can you count every order that has the required number of correct answers?

1. \(X\sim\operatorname{Bin}(15,0.25)\). 2. \(P(X=5)=\binom{15}{5}(0.25)^5(0.75)^{10}\). 3. Therefore, \(P(X=5)\approx0.1651\).

\(\binom{15}{5}(0.25)^5(0.75)^{10}\approx0.1651\).

- Find the probability of one color on a spinner with three equal sections. - For red or blue, combine the probabilities of the two disjoint outcomes. - Use a binomial coefficient to count the arrangements of the successes. - Check that the success and failure probabilities add to \(1\).

1. For part a), define success as landing on red. Then \(p=\frac13\), so \(P(X=3)=\binom83\left(\frac13\right)^3\left(\frac23\right)^5=\frac{1792}{6561}\approx0.2731\). 2. For part b), define success as landing on red or blue. Then \(p=\frac23\), so \(P(Y=6)=\binom86\left(\frac23\right)^6\left(\frac13\right)^2=\frac{1792}{6561}\approx0.2731\).

a) \(\frac{1792}{6561}\approx0.2731\), or about \(27.31\%\) b) \(\frac{1792}{6561}\approx0.2731\), or about \(27.31\%\)

- Decide whether the order matters in each part. - Count the ways to place two hits among four shots. - Use the binomial probability formula when only the number of hits is specified. - For one specified sequence, multiply the probabilities along that sequence.

1. The parameters are \(n=4\), \(p=0.8\), and \(1-p=0.2\). 2. For part a), \(P(X=2)=\binom42(0.8)^2(0.2)^2=6\cdot0.64\cdot0.04=0.1536\). 3. For part b), the order is fixed as hit, miss, miss, hit. Its probability is \(0.8\cdot0.2\cdot0.2\cdot0.8=(0.8)^2\cdot(0.2)^2=0.0256\).

a) \(0.1536\), or \(15.36\%\) b) \(0.0256\), or \(2.56\%\)

- A binomial experiment requires independent trials and a constant success probability. - How does sampling without replacement affect the next trial? - When the population is effectively very large, can the probability be modeled as constant? - Does one die roll affect the next roll?

1. In a), the selections are without replacement from a small population. The probability of selecting a defective screw changes after each selection, and the trials are dependent. This is not a binomial experiment. 2. In b), the model assumes independent trials and a constant defect probability. This is a binomial experiment with \(n=100\) and \(p=0.005\). 3. In c), the die rolls are independent, and the probability of an even result is constant: \(p=\frac{3}{6}=0.5\). This is a binomial experiment with \(n=8\) and \(p=0.5\).

a) Not binomial; without replacement, the success probability changes and the trials are dependent. b) Binomial with \(n=100\) and \(p=0.005\). c) Binomial with \(n=8\) and \(p=0.5\).

- Check whether there are exactly two relevant outcomes on each trial. - Does the success probability remain constant? - Can the trials be treated as independent?

1. In a), each question has two relevant outcomes, correct or incorrect. The guesses are independent and the success probability is constant at \(p=\frac{1}{4}=0.25\). This is binomial with \(n=10\). 2. In b), students are selected without replacement from a small group. The probability of selecting a girl changes after each selection, so this is not binomial. 3. In c), the spins are independent and the probability of yellow remains \(p=\frac{1}{3}\). This is binomial with \(n=15\).

a) Binomial with \(n=10\) and \(p=0.25\). b) Not binomial because selection is without replacement and the success probability changes. c) Binomial with \(n=15\) and \(p=\frac{1}{3}\).

- Are there several possible positions for the two correct answers? - What event does multiplying two success probabilities and four failure probabilities describe? - Think of all paths in a tree diagram that contain exactly two successes. - Does the order of the two correct answers matter to the event?

1. The test can be modeled as a binomial setting with \(n=6\) and \(p=\frac{1}{3}\). 2. The expression \(\left(\frac{1}{3}\right)^2\left(\frac{2}{3}\right)^4\) gives the probability of one specific answer pattern containing two correct answers and four incorrect answers. 3. There are \(\binom{6}{2}=15\) possible positions for the two correct answers. 4. Therefore, the full probability is \(P(X=2)=\binom{6}{2}\left(\frac{1}{3}\right)^2\left(\frac{2}{3}\right)^4=\frac{80}{243}\approx0.3292\).

The expression represents only one specific arrangement of two correct and four incorrect answers. Because there are \(\binom{6}{2}=15\) such arrangements, it must be multiplied by \(15\). Thus, \(P(X=2)=\frac{80}{243}\approx0.3292\).

- Distinguish a specified sequence from an event defined only by the total number of defects. - When only the first outcomes are specified, later outcomes do not affect the event. - Count the possible positions for a fixed number of defects.

1. The probability of a nondefective component is \(0.9\), and the probability of a defective component is \(0.1\). For the fixed sequence in part 1, \(P=(0.9)^5(0.1)=0.059049\). 2. Let \(X\) count defective components. Then \(X\) is binomial with \(n=6\) and \(p=0.1\). Thus, \(P(X=1)=\binom{6}{1}(0.1)(0.9)^5=0.354294\). 3. Only the first three outcomes are restricted, so the remaining outcomes do not affect the event. The probability is \((0.9)^3=0.729\).

1. \(0.059049\), or about \(5.90\%\) 2. \(0.354294\), or about \(35.43\%\) 3. \(0.729\), or \(72.9\%\)

- Identify \(n\) and \(p\) for the experiment. - Distinguish between a probability for one exact value and a cumulative probability. - Express an interval probability as the difference of two cumulative probabilities. - Use the binomial probability functions on your calculator.

1. The random variable \(X\) has a binomial distribution with \(n=15\) and \(p=\frac{1}{6}\). 2. \(P(X=3)=\binom{15}{3}\left(\frac{1}{6}\right)^3\left(\frac{5}{6}\right)^{12}\approx 0.2363\). 3. A cumulative binomial calculation gives \(P(X\le 4)\approx 0.9102\). 4. \(P(2\le X\le 5)=P(X\le 5)-P(X\le 1)\). 5. Since \(P(X\le 5)\approx 0.9726\) and \(P(X\le 1)\approx 0.2596\), the probability is approximately \(0.7130\).

1. \(P(X=3)\approx 0.2363\) 2. \(P(X\le 4)\approx 0.9102\) 3. \(P(2\le X\le 5)\approx 0.7130\)

- Write \(F(k)\) and \(F(k-1)\) as sums of individual probabilities. - Which outcome is included in \(F(k)\) but not in \(F(k-1)\)? - Identify the terms that cancel when the sums are subtracted.

1. By definition, \(F(k)=\sum_{i=0}^{k}P(X=i)\). 2. Similarly, \(F(k-1)=\sum_{i=0}^{k-1}P(X=i)\). 3. In the difference \(F(k)-F(k-1)\), every probability from \(P(X=0)\) through \(P(X=k-1)\) cancels. 4. The only remaining term is \(P(X=k)\). 5. Therefore, \(F(k)-F(k-1)=P(X=k)\).

Subtracting \(F(k-1)\) from \(F(k)\) removes all probabilities for values below \(k\), leaving exactly \(P(X=k)\).

- Use the summation limits to identify the included values of the random variable. - For an expression beginning with \(1-\), identify the complement of the summed event. - Read \(n\), \(p\), and the possible values directly from each binomial term.

1. Part a) is the cumulative probability for at most \(2\) successes when \(n=15\) and \(p=0.1\). 2. \(P(X\le 2)\approx 0.8159\). 3. In part b), the sum is \(P(X\le 1)\) for \(n=5\) and \(p=0.4\). 4. Therefore, the full expression is \(1-P(X\le 1)=P(X\ge 2)\approx 0.6630\).

a) At most \(2\) successes; \(P\approx 0.8159\) b) At least \(2\) successes; \(P\approx 0.6630\)

- Without a binomial coefficient, is the order fixed or unrestricted? - Interpret the exponents as counts of defective and nondefective chips. - Use the relationship between an event and its complement. - Interpret the binomial coefficient as the number of possible arrangements.

1. Let a defective chip be a success. Then \(n=8\), \(p=0.05\), and \(1-p=0.95\). 2. In part a), there is no binomial coefficient, so the expression represents one specified order containing two defective and six nondefective chips, such as only the first two chips being defective. 3. In part b), \(\binom{8}{2}\) counts all possible positions of the two defective chips, so the expression represents exactly two defective chips in the sample. 4. In part c), \((0.95)^8\) is the probability that no chips are defective. Its complement represents at least one defective chip.

a) One specified sequence with two defective chips and six nondefective chips, such as only the first two chips being defective b) Exactly two of the eight chips are defective c) At least one of the eight chips is defective

- Does a specified order require counting several arrangements? - Use a binomial coefficient when the positions of the makes can vary. - Translate “more than two” into the possible numbers of makes. - Recall the formula for exactly \(k\) successes in a binomial setting.

1. The make probability is \(p=0.6\), and the miss probability is \(0.4\). 2. For part a), the sequence is make, make, miss, miss, so the probability is \((0.6)^2(0.4)^2=0.0576\). 3. For part b), the two makes can occur in any two positions, so \(P(X=2)=\binom{4}{2}(0.6)^2(0.4)^2=0.3456\). 4. For part c), more than two makes means three or four makes. Therefore, \(P(X>2)=\binom{4}{3}(0.6)^3(0.4)+\binom{4}{4}(0.6)^4=0.4752\).

a) \((0.6)^2(0.4)^2=0.0576\) b) \(\binom{4}{2}(0.6)^2(0.4)^2=0.3456\) c) \(\binom{4}{3}(0.6)^3(0.4)+\binom{4}{4}(0.6)^4=0.4752\)

- Write both probabilities using the binomial formula. - Use the symmetry of \(\binom{5}{k}\) and \(\binom{5}{5-k}\). - Compare the powers in the two products.

1. For the first athlete, \(P(X=4)=\binom{5}{4}(0.8)^4(0.2)=0.4096\). 2. For the second athlete, \(P(Y=1)=\binom{5}{1}(0.2)(0.8)^4=0.4096\). 3. In general, \(P(X=k)=\binom{5}{k}(0.8)^k(0.2)^{5-k}\). 4. Also, \(P(Y=5-k)=\binom{5}{5-k}(0.2)^{5-k}(0.8)^k\). 5. Since \(\binom{5}{5-k}=\binom{5}{k}\), the two probabilities are equal.

a) \(P(X=4)=0.4096\) and \(P(Y=1)=0.4096\) b) The equality follows from \(\binom{5}{k}=\binom{5}{5-k}\) and the fact that the same factors \((0.8)^k\) and \((0.2)^{5-k}\) appear in both probabilities.

- Is the probability of success the same on every trial? - Does replacement matter in each situation? - Is the number of trials fixed in advance? - Compare the sample size with the population size.

1. In a), the candies are selected without replacement from a small population. The probability of selecting a red candy changes during the selection, so \(X\) is not binomial. 2. In b), each spin has two relevant outcomes, red or not red, with constant probability \(p=\frac{1}{3}\). If the spins are independent, then \(X\sim\operatorname{Bin}(10,\frac{1}{3})\). 3. In c), sampling is technically without replacement, but \(50\) is extremely small compared with \(500{,}000\). If households can be treated as approximately independent and the ownership rate is stable at \(0.12\), a binomial model with \(n=50\) and \(p=0.12\) is a reasonable approximation.

a) Not binomial; the success probability changes in a small sample without replacement. b) Binomial if spins are independent, with \(n=10\) and \(p=\frac{1}{3}\). c) Approximately binomial with \(n=50\) and \(p=0.12\), assuming approximate independence and a stable population rate.

- Determine the smallest count that is a majority of \(15\). - Define a binomial count for regular bicycle commuters. - Use a cumulative probability or its complement.

1. Let \(X\) be the number of people in the group who regularly commute by bicycle. Then \(X\sim\operatorname{Bin}(15,0.60)\). 2. A majority of \(15\) people means at least \(8\) people. 3. Therefore, \(P(X\ge8)=1-P(X\le7)\approx0.7869.\)

\(P(X\ge8)\approx0.7869\).

- Recall all four binomial conditions, including the fixed number of trials. - Define a success as a misrouted package. - Match the sample size and single-package probability to the binomial parameters.

1. A binomial model requires a fixed number of trials, two relevant outcomes on each trial, independent trials, and a constant success probability. 2. Here the fixed number is \(40\), the two outcomes are misrouted or not misrouted, independence must be reasonable for the sampled packages, and the misrouting probability must remain \(0.05\). 3. Let \(X\) be the number of misrouted packages among the \(40\) sampled packages. Then \(n=40\) and \(p=0.05\). 4. \(P(X=2)=\binom{40}{2}(0.05)^2(0.95)^{38}\approx0.2777.\)

a) Fixed \(n=40\), two relevant outcomes, independence, and constant \(p=0.05\). b) \(X\) is the number of misrouted packages; \(n=40\), \(p=0.05\). c) \(P(X=2)\approx0.2777\).

- Identify the binomial parameters from the context. - Express the inclusive interval in \(F\) as a difference of cumulative probabilities. - Find the complement of the event \(X\ge175\).

1. \(X\sim\operatorname{Bin}(200,0.80)\). 2. \(P(E)=P(X=165)=\binom{200}{165}(0.80)^{165}(0.20)^{35}\approx0.0495.\) 3. \(P(F)=P(155\le X\le170)=P(X\le170)-P(X\le154)\approx0.8066.\) 4. The complement of \(X\ge175\) is \(X\le174\). Therefore, \(1-P(X\ge175)\) is the probability that at most \(174\) of the \(200\) inspected trees are pine trees.

a) \(P(E)\approx0.0495\) and \(P(F)\approx0.8066\). b) It is the probability that at most \(174\) of the \(200\) inspected trees are pine trees.

- What is the probability of a correct answer when guessing among \(4\) choices? - Identify the total number of trials. - “At most” includes all values from \(0\) through the stated number. - For “at least,” a complement can reduce the number of terms.

1. Let \(X\) be the number of correct answers. Then \(X\) has a binomial distribution with \(n=15\) and \(p=0.25\). 2. For part a), \(P(X=6)=\binom{15}{6}(0.25)^6(0.75)^9\). 3. For part b), \(P(X\le 1)=(0.75)^{15}+15(0.25)(0.75)^{14}\). 4. For part c), use the complement: \(P(X\ge 3)=1-\sum_{k=0}^{2}\binom{15}{k}(0.25)^k(0.75)^{15-k}\).

a) \(\binom{15}{6}(0.25)^6(0.75)^9\) b) \((0.75)^{15}+15(0.25)(0.75)^{14}\) c) \(1-\sum_{k=0}^{2}\binom{15}{k}(0.25)^k(0.75)^{15-k}\)

- Convert “1 in 20” to a probability. - Use \(np\) for the expected count. - Use a point probability for exactly one defect and a cumulative probability for at most two.

1. The defect probability is \(p=\frac{1}{20}=0.05\). 2. For part a), the expected number is \(E(X)=1400(0.05)=70.\) 3. For parts b) and c), let \(X\sim\operatorname{Bin}(25,0.05)\). 4. \(P(X=1)=\binom{25}{1}(0.05)(0.95)^{24}\approx0.3650.\) 5. \(P(X\le2)=\sum_{k=0}^{2}\binom{25}{k}(0.05)^k(0.95)^{25-k}\approx0.8729.\)

a) \(70\) defective diodes. b) \(0.3650\). c) \(0.8729\).

- State all four binomial conditions, including the fixed number of trials. - Identify \(n\) and \(p\). - Use the binomial-mode rule involving \((n+1)p\).

1. A binomial model requires a fixed number of trials, two relevant outcomes on each trial, independent trials, and a constant success probability. 2. Here, the fixed number is \(n=45\), the two outcomes are make or miss, the shots must be independent, and the make probability must remain \(p=0.75\). 3. Since \((n+1)p=46(0.75)=34.5\) is not an integer, the unique mode is \(\lfloor34.5\rfloor=34.\)

a) Fixed \(n=45\), two relevant outcomes, independence, and constant \(p=0.75\). b) \(34\) made free throws.

- Which measure describes the center of a binomial distribution? - Near which value would you expect the tallest bar of the distribution? - Recall the relationship between the mode and the parameters of a binomial distribution.

1. For a binomial distribution, when \((n+1)p\) is not an integer, the unique mode is \(\lfloor(n+1)p\rfloor\). 2. Here, \((80+1)\cdot0.45=81\cdot0.45=36.45\), which is not an integer. 3. Therefore, the mode is \(\lfloor36.45\rfloor=36\), so \(P(X=36)\) is the greatest probability.

The greatest probability occurs at \(k=36\).

- How many relevant outcomes are recorded on each trial? - Does the success probability stay the same? - Does one trial affect the following trials?

1. In a), there are two relevant outcomes, the success probability is constant, and the trials are independent. This is binomial with \(n=12\) and \(p=0.80\). 2. In b), drawing without replacement changes the probability of an ace after each draw. For example, after the first draw the next probability is either \(\frac{3}{51}\) or \(\frac{4}{51}\). The trials are dependent, so the experiment is not binomial.

a) Binomial with \(n=12\) and \(p=0.80\). b) Not binomial because the ace probability changes without replacement.

- Can each trial be divided into exactly two relevant categories? - How many repetitions are made? - What is the success probability on one trial, and does it stay constant?

1. In a), each roll is classified as prime or not prime. The prime faces are \(2\), \(3\), and \(5\), so \(p=\frac{3}{6}=0.5\). The rolls are independent, so this is binomial with \(n=20\). 2. In b), the recorded outcome has \(10\) categories rather than two relevant categories. As stated, this is not a binomial experiment.

a) Binomial with \(n=20\) and \(p=0.5\). b) Not binomial because the recorded outcome has more than two categories.

- Use one binomial model for all three events in part a). - At least four and at most three are complementary events. - Independence means previous outcomes do not change the next-trial probability.

1. Let \(X\sim\operatorname{Bin}(50,0.04)\). 2. \(P(X=2)\approx0.2762.\) 3. \(P(X\ge4)\approx0.1391.\) 4. \(P(X\le3)\approx0.8609.\) 5. Independence means the first \(100\) results do not change the next-component defect probability, so \(P(\text{next defective})=0.04.\)

a) (1) \(0.2762\); (2) \(0.1391\); (3) \(0.8609\). b) \(0.04\).

- Translate the stated range into an inequality for the binomial count. - Use a difference of cumulative probabilities. - Check carefully which values fall below the inclusive lower boundary before subtracting.

1. Let \(X\sim\operatorname{Bin}(120,0.15)\). 2. The required event is \(12\le X\le24\). 3. \(P(12\le X\le24)=P(X\le24)-P(X\le11)\approx0.9052\).

\(0.9052\).

- What conditions must a binomial experiment satisfy? - Does the probability of success change from trial to trial? - How does replacement affect independence? - Are there real-world factors that might challenge a constant success probability?

1. In a), replacement keeps the red-ball probability constant at \(p=\frac{8}{40}=0.20\), and the draws are independent. The model is binomial with \(n=5\), and \(X\) is the number of red balls drawn. 2. In b), under the modeling assumptions that the success probability remains constant and the shots are independent, the model is binomial with \(n=10\) and \(p=0.75\). The random variable \(X\) is the number of made free throws. 3. In c), drawing without replacement changes the defect probability on the second draw. The trials are dependent, so the model is not binomial.

a) Yes: \(n=5\), \(p=0.20\), and \(X\) is the number of red balls drawn. b) Yes, assuming independence and constant probability: \(n=10\), \(p=0.75\), and \(X\) is the number of made shots. c) No; without replacement, the success probability changes.

- What conditions must hold for a sequence of Bernoulli trials? - Does each seed have the same germination probability? - Use the binomial probability formula for an exact number of successes. - “At most \(1\)” includes \(0\) and \(1\) success.

1. Each trial has two outcomes, germination or no germination. The probability of germination is constant at \(0.80\), and the outcomes are independent. Therefore, the conditions for Bernoulli trials are met. 2. Let \(X\sim\operatorname{Bin}(6,0.80)\). Then \(P(X=5)=\binom{6}{5}(0.80)^5(0.20)=0.393216\). 3. For part c), \(P(X\le 1)=P(X=0)+P(X=1)=(0.20)^6+6(0.80)(0.20)^5=0.0016\).

a) There are two outcomes per seed, the success probability is constant, and the outcomes are independent. b) The probability is \(0.393216\), or approximately \(39.32\%\). c) The probability is \(0.0016\), or \(0.16\%\).

- State all four Bernoulli/binomial conditions. - Use the binomial point formula for exactly one success. - Use a complement for at least two successes.

1. Each bottle must be classified into two relevant outcomes, the sample size must be fixed at \(10\), the air-bubble probability must remain \(0.04\), and the outcomes must be independent. 2. Let \(X\sim\operatorname{Bin}(10,0.04)\). Then \(P(X=1)=\binom{10}{1}(0.04)(0.96)^9\approx0.2770.\) 3. \(P(X\ge2)=1-\left[(0.96)^{10}+10(0.04)(0.96)^9\right]\approx0.0582.\)

a) Fixed \(n=10\), two relevant outcomes, constant \(p=0.04\), and independence. b) \(0.2770\). c) \(0.0582\).

- What count of red outcomes is required by “exactly once”? - How are the probabilities of red and not red related? - For “at least once,” consider the complement event. - Compare your result with \(90\%\).

1. Let \(X\sim\operatorname{Bin}(4,0.25)\). 2. For part a), \(P(X=1)=\binom{4}{1}(0.25)(0.75)^3=0.421875\). 3. For part b), \(P(X\ge 1)=1-P(X=0)=1-(0.75)^4=0.68359375\). 4. Since \(68.36\%<90\%\), the claim is false.

a) The probability is \(0.421875\), or \(42.1875\%\). b) The claim is false. The probability of red at least once is approximately \(68.36\%\), not more than \(90\%\).

- Find the probability of guessing one question correctly. - Identify the possible values of \(X\). - Use the formula for exactly \(k\) successes in a fixed number of independent trials. - Check that the unrounded probabilities add to \(1\).

1. The number of correct answers is binomial with \(n=5\), \(p=\frac{1}{3}\), and \(1-p=\frac{2}{3}\). 2. Apply \(P(X=k)=\binom{5}{k}\left(\frac{1}{3}\right)^k\left(\frac{2}{3}\right)^{5-k}\) for \(k\in\{0,1,2,3,4,5\}\). 3. The probabilities are \(P(X=0)=\frac{32}{243}\approx0.1317\), \(P(X=1)=\frac{80}{243}\approx0.3292\), \(P(X=2)=\frac{80}{243}\approx0.3292\), \(P(X=3)=\frac{40}{243}\approx0.1646\), \(P(X=4)=\frac{10}{243}\approx0.0412\), and \(P(X=5)=\frac{1}{243}\approx0.0041\).

<table> <tr><td>\(k\)</td><td>\(0\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td><td>\(4\)</td><td>\(5\)</td></tr> <tr><td>\(P(X=k)\)</td><td>\(0.1317\)</td><td>\(0.3292\)</td><td>\(0.3292\)</td><td>\(0.1646\)</td><td>\(0.0412\)</td><td>\(0.0041\)</td></tr> </table>

- First determine the probability of one specified color on a single spin. - Which possible counts are included in “at least \(5\)”? - Model each specified color as a success in a sequence of Bernoulli trials. - Add the probabilities of the required cases.

1. For either specified color, the success probability is \(p=\frac{1}{3}\). 2. For part 1, \(P(X=2)=\binom{6}{2}\left(\frac{1}{3}\right)^2\left(\frac{2}{3}\right)^4=\frac{80}{243}\approx 0.329218\). 3. For part 2, \(P(X\ge 5)=P(X=5)+P(X=6)\). 4. Therefore, \(P(X\ge 5)=\binom{6}{5}\left(\frac{1}{3}\right)^5\left(\frac{2}{3}\right)+\left(\frac{1}{3}\right)^6=\frac{13}{729}\approx 0.017833\).

1. The probability is \(\frac{80}{243}\approx 32.92\%\). 2. The probability is \(\frac{13}{729}\approx 1.78\%\).

- Decide what one success means before writing the binomial model. - Which outcome sequence gives no makes at all? - For each tail, check carefully whether the boundary count is included.

1. Let \(X\sim\operatorname{Bin}(6,0.70)\). 2. \(P(X=4)=\binom{6}{4}(0.70)^4(0.30)^2=0.324135\approx0.3241\). 3. \(P(X=0)=(0.30)^6=0.000729\approx0.0007\). 4. \(P(X\le1)=P(X=0)+P(X=1)=0.010935\approx0.0109\). 5. \(P(X\ge5)=P(X=5)+P(X=6)=0.420175\approx0.4202\).

a) \(0.3241\). b) \(0.0007\). c) \(0.0109\). d) \(0.4202\).

- Let success mean that a bulb is defective. - Use the binomial point-probability formula for exactly two defects. - For “at least \(1\),” use the complement of no defects.

1. Let \(X\sim\operatorname{Bin}(8,0.05)\), where \(X\) is the number of defective bulbs. 2. \(P(X=0)=(0.95)^8\approx0.6634\). 3. \(P(X=2)=\binom{8}{2}(0.05)^2(0.95)^6\approx0.0515\). 4. \(P(X\ge1)=1-P(X=0)\approx0.3366\).

a) \(0.6634\). b) \(0.0515\). c) \(0.3366\).

- How many of the six die faces count as a success? - Make sure the lower-tail event includes its boundary count. - Use a complement for “at least \(3\).”

1. Let \(X\sim\operatorname{Bin}(10,\frac{1}{6})\). 2. \(P(X=2)=\binom{10}{2}(\frac{1}{6})^2(\frac{5}{6})^8\approx0.2907\). 3. \(P(X\le1)=(\frac{5}{6})^{10}+10(\frac{1}{6})(\frac{5}{6})^9\approx0.4845\). 4. \(P(X\ge3)=1-P(X\le2)\approx0.2248\).

a) \(0.2907\). b) \(0.4845\). c) \(0.2248\).

- Sampling with replacement keeps the success probability constant and supports independence. - “No defects or exactly one defect” is the event \(X\le1\). - Add the two binomial point probabilities.

1. Let \(X\sim\operatorname{Bin}(15,0.10)\). 2. The required event is \(X\le1\), so \(P(X\le1)=(0.90)^{15}+15(0.10)(0.90)^{14}.\) 3. Therefore, \(P(X\le1)\approx0.5490.\)

\(0.5490\).

- Determine the one-spin prize probability from the equal sections. - The complement of “at least \(3\)” contains \(0\), \(1\), and \(2\) successes. - Use the binomial distribution for the ten independent spins.

1. The success probability is \(p=\frac{1}{5}=0.20\), so \(X\sim\operatorname{Bin}(10,0.20)\). 2. Use the complement: \(P(X\ge3)=1-P(X\le2)\). 3. Thus, \(P(X\ge3)=1-\sum_{k=0}^{2}\binom{10}{k}(0.20)^k(0.80)^{10-k}\approx0.3222\).

\(P(X\ge3)\approx0.3222\).

- Sampling with replacement keeps the red probability constant from draw to draw. - Translate “at most \(2\)” into the included success counts. - Add the corresponding binomial point probabilities.

1. Let \(X\sim\operatorname{Bin}(15,0.20)\). 2. “At most \(2\)” means \(X=0,1,2\). 3. Therefore, \(P(X\le2)=\sum_{k=0}^{2}\binom{15}{k}(0.20)^k(0.80)^{15-k}\approx0.3980\).

\(P(X\le2)\approx0.3980\).

- Determine the minimum number of games Sarah must win. - Model the number of wins with a binomial random variable. - Add the probabilities of all winning counts.

1. Let \(X\sim\operatorname{Bin}(7,0.55)\), where \(X\) is the number of games Sarah wins. 2. Sarah wins the match when \(X\ge4\). 3. \(P(X\ge4)=\sum_{k=4}^{7}\binom{7}{k}(0.55)^k(0.45)^{7-k}\approx0.6083.\)

\(0.6083\).

- Determine the smallest number of wins that is more than half of ten games. - Model Marc's number of wins with a binomial random variable. - Add the probabilities of all series-winning counts.

1. Let \(X\sim\operatorname{Bin}(10,0.40)\), where \(X\) is the number of games Marc wins. 2. Marc is the overall winner exactly when \(X\ge6\). 3. Therefore, \(P(X\ge6)=\sum_{k=6}^{10}\binom{10}{k}(0.40)^k(0.60)^{10-k}\approx0.1662\).

\(P(\text{Marc wins the series})\approx0.1662\).

- Identify \(n=15\) and \(p=0.40\). - Use one point probability for part a). - Translate the “at most” and “at least” phrases into the included integer counts.

1. Let \(X\sim\operatorname{Bin}(15,0.40)\). 2. \(P(X=6)=\binom{15}{6}(0.40)^6(0.60)^9\approx0.2066\). 3. \(P(X\le2)=\sum_{k=0}^{2}\binom{15}{k}(0.40)^k(0.60)^{15-k}\approx0.0271\). 4. \(P(X\ge13)=\sum_{k=13}^{15}\binom{15}{k}(0.40)^k(0.60)^{15-k}\approx0.0003\).

a) \(0.2066\). b) \(0.0271\). c) \(0.0003\).

- Let success mean that a component is defective. - A complement is efficient for “at least \(2\).” - “No components are defective” means \(X=0\).

1. Let \(X\sim\operatorname{Bin}(15,0.04)\). 2. \(P(X=1)=\binom{15}{1}(0.04)(0.96)^{14}\approx0.3388\). 3. \(P(X\ge2)=1-[P(X=0)+P(X=1)]\approx0.1191\). 4. \(P(X=0)=(0.96)^{15}\approx0.5421\).

a) \(0.3388\). b) \(0.1191\). c) \(0.5421\).

- Use the given success probability to define a binomial count. - A complement is efficient for “at least \(2\).” - List the included counts for the inclusive interval before calculating.

1. \(X\sim\operatorname{Bin}(10,0.40)\). 2. \(P(X=4)=\binom{10}{4}(0.40)^4(0.60)^6\approx0.2508\). 3. \(P(X\ge2)=1-P(X\le1)\approx0.9536\). 4. \(P(3\le X\le5)=P(X=3)+P(X=4)+P(X=5)\approx0.6665\).

a) \(0.2508\). b) \(0.9536\). c) \(0.6665\).

- Identify the binomial parameters from the sample size and defect rate. - Use a complement for “more than \(2\).” - For part c), list the included integer counts before calculating.

1. \(X\sim\operatorname{Bin}(15,0.05)\). 2. \(P(X=1)=\binom{15}{1}(0.05)(0.95)^{14}\approx0.3658\). 3. \(P(X>2)=1-P(X\le2)\approx0.0362\). 4. \(P(1\le X\le3)=P(X=1)+P(X=2)+P(X=3)\approx0.5312\).

a) \(0.3658\). b) \(0.0362\). c) \(0.5312\).

- Translate the verbal condition into the included win counts. - Use the binomial formula for each exact count. - Add the three disjoint point probabilities.

1. The number of wins has distribution \(X\sim\operatorname{Bin}(10,0.20).\) 2. The required event is \(1\le X\le3.\) 3. Therefore, \(P(1\le X\le3) =\sum_{k=1}^{3}\binom{10}{k}(0.20)^k(0.80)^{10-k} \approx0.7718.\)

\(P(1\le X\le3) =\sum_{k=1}^{3}\binom{10}{k}(0.20)^k(0.80)^{10-k} \approx0.7718.\)

- Identify the three included integer success counts. - Use the binomial point-probability formula for each count. - Add the probabilities because the exact-count events are disjoint.

1. The number made has distribution \(X\sim\operatorname{Bin}(15,0.60).\) 2. The required event is \(8\le X\le10.\) 3. Therefore, \(P(8\le X\le10) =\sum_{k=8}^{10}\binom{15}{k}(0.60)^k(0.40)^{15-k} \approx0.5696.\)

\(P(8\le X\le10) =\sum_{k=8}^{10}\binom{15}{k}(0.60)^k(0.40)^{15-k} \approx0.5696.\)

- Convert the no-show probability to an attendance probability. - Determine which attendance counts exceed the available seats. - Add the corresponding binomial point probabilities.

1. Each person attends with probability \(0.90\), so \(X\sim\operatorname{Bin}(15,0.90)\). 2. The theater is over capacity when \(X>13\), so \(X=14\) or \(X=15\). 3. Therefore, \(P(X>13)=\binom{15}{14}(0.90)^{14}(0.10)+(0.90)^{15}\approx0.5490\).

\(P(\text{over capacity})\approx0.5490\).

- Convert the cancellation probability to the probability that a reservation is used. - Identify the attendance counts that would create a shortage. - Use the complement of the shortage event.

1. Each reservation is used with probability \(0.75\), so \(X\sim\operatorname{Bin}(12,0.75)\). 2. There are enough tables when \(X\le10\). 3. Using the complement, \(P(X\le10)=1-[P(X=11)+P(X=12)]\approx0.8416\).

\(P(\text{enough tables})\approx0.8416\).

- Model the number of affected vehicles with a binomial distribution. - The greatest probability occurs near the mean. - Use \((n+1)p\) to locate the mode. - Compare the nearby integer probabilities as a check.

1. Let \(X\) be the number of vehicles with a paint defect. Then \(X\) is binomial with \(n=30\) and \(p=0.08\). 2. Since \((n+1)p=31\cdot0.08=2.48\) is not an integer, the unique mode is \(\lfloor2.48\rfloor=2\). 3. As a check, \(P(X=2)\approx0.2696\), while \(P(X=3)\approx0.2188\).

\(2\) vehicles with a paint defect

- Model germination as independent success-or-failure trials. - Calculate \((n+1)p\). - Recall what happens when \((n+1)p\) is an integer. - Check the two adjacent values near the mean.

1. Let \(X\) be the number of seeds that germinate. Then \(X\) is binomial with \(n=15\) and \(p=0.75\). 2. \((n+1)p=16\cdot0.75=12\), which is an integer. 3. When \((n+1)p\) is an integer, the two modes are that integer and one less. Therefore, the modes are \(12\) and \(11\). 4. Indeed, \(P(X=11)=P(X=12)\approx0.2252\).

\(11\) and \(12\) germinating seeds

- Define success for one draw. - Explain why replacement keeps the success probability unchanged. - Count the possible positions of the one successful draw.

1. Define success as drawing the ball labeled \(2\). Because the ball is replaced after each draw, the trials are independent with \(n=3\) and \(p=\frac{1}{5}=0.2\). 2. The probability of exactly one success is \(P(X=1)=\binom{3}{1}(0.2)(0.8)^2\). 3. Therefore, \(P(X=1)=3\cdot0.2\cdot0.64=0.384\).

\(P(X=1)=0.384\), or \(38.4\%\)

- Model the number of made free throws with a binomial random variable. - Use the point-probability formula for exactly three makes. - Think about which whole-number counts meet the “at least” condition.

1. Let \(X\sim\operatorname{Bin}(5,0.70)\). 2. \(P(X=3)=\binom{5}{3}(0.70)^3(0.30)^2=0.3087\). 3. \(P(X\ge4)=P(X=4)+P(X=5)\). 4. \(P(X=4)=\binom{5}{4}(0.70)^4(0.30)=0.36015\), and \(P(X=5)=(0.70)^5=0.16807\). 5. Therefore, \(P(X\ge4)=0.52822\).

a) \(0.3087\). b) \(0.52822\).

- What fraction of the spinner's sections count as blue? - “At most \(1\)” includes \(0\) and \(1\). - Use the complement of zero blue results for part c).

1. \(X\sim\operatorname{Bin}(6,0.20)\). 2. \(P(X=2)=\binom{6}{2}(0.20)^2(0.80)^4=0.24576\). 3. \(P(X\le1)=P(X=0)+P(X=1)=0.65536\). 4. \(P(X\ge1)=1-P(X=0)=1-(0.80)^6=0.737856\).

a) \(0.24576\). b) \(0.65536\). c) \(0.737856\).

- Use the sample size and defect probability as the binomial parameters. - For “more than \(2\),” use the complement through \(2\). - Keep unrounded values through the final subtraction.

1. \(X\sim\operatorname{Bin}(10,0.05)\). 2. \(P(X=1)=\binom{10}{1}(0.05)(0.95)^9\approx0.3151\). 3. \(P(X=0)=(0.95)^{10}\approx0.5987\). 4. \(P(X>2)=1-P(X\le2)\approx0.0115\), using unrounded probabilities in the subtraction.

a) \(0.3151\). b) \(0.5987\). c) \(0.0115\).

- Use the binomial model with \(n=8\) and \(p=\frac{1}{4}\). - Which possible values of \(X\) satisfy the strict inequality \(X>6\)? - Keep an exact form for the very small upper-tail probability before rounding.

1. Since a random guess is correct with probability \(0.25\), \(X\sim\operatorname{Bin}(8,0.25)\). 2. \(P(X=3)=\binom{8}{3}(0.25)^3(0.75)^5\approx0.2076\). 3. More than \(6\) means \(7\) or \(8\) correct answers. 4. Thus, \(P(X>6)=P(X=7)+P(X=8)=\frac{25}{65536}\approx0.000381\).

a) \(0.2076\). b) \(\frac{25}{65536}\approx0.000381\).

- Identify the number of trials and the sprouting probability. - How do successes and failures fit into the required count? - Use the binomial point-probability formula.

1. Let \(X\sim\operatorname{Bin}(12,0.95)\). 2. \(P(X=11)=\binom{12}{11}(0.95)^{11}(0.05)\). 3. Therefore, \(P(X=11)\approx0.3413\).

\(\binom{12}{11}(0.95)^{11}(0.05)\approx0.3413\).

- Let success mean a made free throw. - How can you find the number of misses from the number of makes? - Use the binomial point-probability formula.

1. Let \(X\sim\operatorname{Bin}(10,0.70)\). 2. \(P(X=7)=\binom{10}{7}(0.70)^7(0.30)^3\). 3. Therefore, \(P(X=7)\approx0.2668\).

\(\binom{10}{7}(0.70)^7(0.30)^3\approx0.2668\).

- Define success as a correct guess. - State the fixed number of questions and the one-question success probability. - “No correct answers” corresponds to the binomial count \(X=0\).

1. Let \(X\) be the number of correct answers. Each guess is correct with probability \(1/4\), so \(X\sim\operatorname{Bin}(15,0.25)\). 2. No correct answers means \(X=0\). 3. Thus, \(P(X=0)=\binom{15}{0}(0.25)^0(0.75)^{15}=(0.75)^{15}\approx0.01336\).

a) \(X\sim\operatorname{Bin}(15,0.25)\). b) \((0.75)^{15}\approx0.01336\).

- Define success as arrival within \(24\) hours. - State \(n\) and \(p\) before calculating. - “All \(20\)” corresponds to the endpoint count \(X=20\).

1. Let \(X\) be the number of packages that arrive within \(24\) hours. Then \(X\sim\operatorname{Bin}(20,0.95)\). 2. All packages arrive within \(24\) hours exactly when \(X=20\). 3. Therefore, \(P(X=20)=\binom{20}{20}(0.95)^{20}(0.05)^0=(0.95)^{20}\approx0.3585\).

a) \(X\sim\operatorname{Bin}(20,0.95)\). b) \((0.95)^{20}\approx0.3585\).

- For one guess, how many of the four choices would earn a correct answer? - What number of correct answers would be exactly half of the test? - Think about what a binomial mean counts over repeated trials.

1. Let \(X\) be the number of correct answers. Then \(X\sim\operatorname{Bin}(10,0.25)\). 2. Exactly half means \(X=5\), so \(P(X=5)=\binom{10}{5}(0.25)^5(0.75)^5\approx0.0584\). 3. The expected number is \(E(X)=np=10(0.25)=2.5\).

a) \(X\) is the number of correct answers, with \(X\sim\operatorname{Bin}(10,0.25)\). b) \(0.0584\). c) \(2.5\) correct answers.

- Use the binomial point formula for exactly three coupons. - For at least one coupon, use the complement of zero coupons. - Keep the same \(n\) and \(p\) in both parts.

1. \(X\sim\operatorname{Bin}(12,0.20)\). 2. \(P(X=3)=\binom{12}{3}(0.20)^3(0.80)^9\approx0.2362\). 3. \(P(X\ge1)=1-P(X=0)=1-(0.80)^{12}\approx0.9313\).

a) \(0.2362\). b) \(0.9313\).

- Use the same binomial model in both parts. - Decide which whole-number counts satisfy “at least” without dropping the boundary value. - Apply the point-probability formula to the included counts.

1. \(X\sim\operatorname{Bin}(6,0.75)\). 2. \(P(X=4)=\binom{6}{4}(0.75)^4(0.25)^2\approx0.2966\). 3. \(P(X\ge5)=\binom{6}{5}(0.75)^5(0.25)+(0.75)^6\approx0.5339\).

a) \(0.2966\). b) \(0.5339\).

- What distinguishes a Bernoulli trial from a more general random experiment? - How many relevant outcome categories does a Bernoulli trial use? - How can the eight numbers be separated into success and failure sets? - Is there only one valid way to define success?

1. One possible definition is “success means an even number.” The success set is \(\{2,4,6,8\}\), so \(p=\frac{4}{8}=0.5\) and \(q=1-p=0.5\). 2. Another possible definition is “success means a number greater than \(6\).” The success set is \(\{7,8\}\), so \(p'=\frac{2}{8}=0.25\).

a) Example: success is an even number. Then \(p=0.5\) and \(q=0.5\). b) Example: success is a number greater than \(6\). Then \(p'=0.25\).

- Decide whether the setting has repeated trials with two outcomes. - Identify the possible values of the number of defective chips. - Record the number of trials and the defect probability. - Apply the binomial probability formula to each possible value. - Check that the unrounded probabilities add to \(1\).

1. Model \(X\) as binomial with \(n=4\), \(p=0.12\), and \(1-p=0.88\). 2. Use \(P(X=k)=\binom{4}{k}(0.12)^k(0.88)^{4-k}\) for \(k\in\{0,1,2,3,4\}\). 3. The probabilities are \(P(X=0)=0.59969536\approx0.5997\), \(P(X=1)=0.32710656\approx0.3271\), \(P(X=2)=0.06690816\approx0.0669\), \(P(X=3)=0.00608256\approx0.0061\), and \(P(X=4)=0.00020736\approx0.0002\).

<table> <tr><th>\(k\)</th><th>\(0\)</th><th>\(1\)</th><th>\(2\)</th><th>\(3\)</th><th>\(4\)</th></tr> <tr><th>\(P(X=k)\)</th><td>\(0.5997\)</td><td>\(0.3271\)</td><td>\(0.0669\)</td><td>\(0.0061\)</td><td>\(0.0002\)</td></tr> </table>

- Define what counts as a success and identify its probability. - Identify the number of trials. - Pair each possible value of \(Y\) with its binomial probability. - Use the binomial coefficient to count arrangements with exactly \(k\) successes.

1. Model \(Y\) as binomial with \(n=5\), \(p=0.28\), and \(1-p=0.72\). 2. Use \(P(Y=k)=\binom{5}{k}(0.28)^k(0.72)^{5-k}\) for \(k\in\{0,1,2,3,4,5\}\). 3. The probabilities are \(P(Y=0)\approx0.1935\), \(P(Y=1)\approx0.3762\), \(P(Y=2)\approx0.2926\), \(P(Y=3)\approx0.1138\), \(P(Y=4)\approx0.0221\), and \(P(Y=5)\approx0.0017\).

<table> <tr><th>\(k\)</th><th>\(0\)</th><th>\(1\)</th><th>\(2\)</th><th>\(3\)</th><th>\(4\)</th><th>\(5\)</th></tr> <tr><th>\(P(Y=k)\)</th><td>\(0.1935\)</td><td>\(0.3762\)</td><td>\(0.2926\)</td><td>\(0.1138\)</td><td>\(0.0221\)</td><td>\(0.0017\)</td></tr> </table>

- State all four binomial conditions. - Identify the fixed trial count and the two relevant outcomes. - Apply the binomial formula to every possible value of \(X\). - Check that the probabilities add to \(1\).

1. A binomial model requires a fixed number of trials, two relevant outcomes on each trial, independent trials, and a constant success probability. 2. Here, \(n=4\), the relevant outcomes are center hit or not, the shots must be independent, and \(p=0.70\) must remain constant. 3. Let \(X\sim\operatorname{Bin}(4,0.70)\). 4. The probabilities are \(P(X=0)=0.0081,\quad P(X=1)=0.0756,\quad P(X=2)=0.2646,\quad P(X=3)=0.4116,\quad P(X=4)=0.2401.\)

a) Fixed \(n=4\), two relevant outcomes, independence, and constant \(p=0.70\). b) \(0.0081,\ 0.0756,\ 0.2646,\ 0.4116,\ 0.2401\) for \(X=0,1,2,3,4\), respectively.

- How would you determine the chance of a red result on one spin? - Complements are efficient for upper or endpoint events. - In part d), use the probability already found in part a) rather than repeating the calculation.

1. \(Y\sim\operatorname{Bin}(12,0.20)\). 2. \(P(Y\ge1)=1-P(Y=0)=1-(0.80)^{12}\approx0.9313\). 3. \(P(Y>3)=1-P(Y\le3)\approx0.2054\). 4. \(P(Y\le10)=1-\left[\binom{12}{11}(0.20)^{11}(0.80)+(0.20)^{12}\right]\approx0.9999998\). 5. The event \(Y=0\) is the complement of \(Y\ge1\), so \(P(Y=0)=1-P(Y\ge1)=(0.80)^{12}\approx0.0687\).

a) \(0.9313\). b) \(0.2054\). c) \(1-\left[\binom{12}{11}(0.20)^{11}(0.80)+(0.20)^{12}\right]\approx0.9999998\). d) \(0.0687\); “no red results” and “at least one red result” are complementary events.

- Use \(n=12\) and \(p=0.15\). - “At most \(2\)” is a lower-tail event. - Use a complement for “more than \(4\).”

1. Let \(X\sim\operatorname{Bin}(12,0.15)\). 2. \(P(X=3)=\binom{12}{3}(0.15)^3(0.85)^9\approx0.1720\). 3. \(P(X\le2)\approx0.7358\). 4. \(P(X>4)=1-P(X\le4)\approx0.0239\).

a) \(0.1720\). b) \(0.7358\). c) \(0.0239\).

- Use \(n=15\) and \(p=0.40\). - Translate “fewer than \(5\)” into an integer inequality. - Use a complement for “at least \(8\).”

1. Let \(X\sim\operatorname{Bin}(15,0.40)\). 2. \(P(X=6)=\binom{15}{6}(0.40)^6(0.60)^9\approx0.2066\). 3. Fewer than five means \(X\le4\), so \(P(X<5)\approx0.2173\). 4. \(P(X\ge8)=1-P(X\le7)\approx0.2131\).

a) \(0.2066\). b) \(0.2173\). c) \(0.2131\).

- Use cumulative binomial probabilities. - Express the upper tail with a complement. - For the interval, subtract the cumulative probability through the value just below the lower endpoint.

1. \(P(X\le4)\approx0.2552\). 2. \(P(X>8)=1-P(X\le8)\approx0.1287\). 3. \(P(5\le X\le10)=P(X\le10)-P(X\le4)\approx0.7192\).

a) \(0.2552\). b) \(0.1287\). c) \(0.7192\).

- Pay particular attention to the strict inequalities in part b). - Convert the open interval to its attainable integer values. - Use a difference of cumulative probabilities.

1. \(P(X\le20)\approx0.0368.\) 2. Because \(X\) is integer-valued, \(22<X<32\) means \(23\le X\le31\). 3. \(P(22<X<32)=P(X\le31)-P(X\le22)\approx0.6980.\)

a) \(0.0368\). b) \(0.6980\).

- Use one binomial model for all three parts. - For the inclusive interval, subtract the cumulative probability through \(7\). - Use a complement for the upper tail.

1. \(X\sim\operatorname{Bin}(200,0.05)\). 2. \(P(X\le10)\approx0.5831\). 3. \(P(8\le X\le12)=P(X\le12)-P(X\le7)\approx0.5832\). 4. \(P(X>15)=1-P(X\le15)\approx0.0444\).

a) \(0.5831\). b) \(0.5832\). c) \(0.0444\).

- Use a complement for the upper-tail event. - “Fewer than \(10\)” ends at \(9\). - Use a difference of cumulative probabilities for the inclusive interval.

1. \(X\sim\operatorname{Bin}(50,0.25)\). 2. \(P(X\ge15)=1-P(X\le14)\approx0.2519\). 3. \(P(X<10)=P(X\le9)\approx0.1637\). 4. \(P(10\le X\le15)=P(X\le15)-P(X\le9)\approx0.6732\).

a) \(0.2519\). b) \(0.1637\). c) \(0.6732\).

- Use complements for upper-tail events. - Use cumulative subtraction for the inclusive interval. - Convert the condition on incorrect answers to a condition on the number correct.

1. \(X\sim\operatorname{Bin}(60,0.25)\). 2. \(P(X>12)=1-P(X\le12)\approx0.7684\). 3. \(P(10\le X\le20)=P(X\le20)-P(X\le9)\approx0.9007\). 4. Fewer than \(40\) incorrect answers means at least \(21\) correct answers, so \(P(X\ge21)=1-P(X\le20)\approx0.0541\).

a) \(0.7684\). b) \(0.9007\). c) \(0.0541\).

- Use one binomial model for all three parts. - Use a complement for “more than \(50\).” - For the inclusive interval, subtract the cumulative probability through \(41\).

1. \(X\sim\operatorname{Bin}(60,0.75)\). 2. \(P(X\le40)\approx0.0925\). 3. \(P(X>50)=1-P(X\le50)\approx0.0452\). 4. \(P(42\le X\le48)=P(X\le48)-P(X\le41)\approx0.7038\).

a) \(0.0925\). b) \(0.0452\). c) \(0.7038\).

- Use a point probability for exactly five defects. - Translate “fewer than \(8\)” to an upper endpoint of \(7\). - Use a complement for the upper tail.

1. \(X\sim\operatorname{Bin}(100,0.05)\). 2. \(P(X=5)=\binom{100}{5}(0.05)^5(0.95)^{95}\approx0.1800\). 3. At least \(3\) but fewer than \(8\) means \(3\le X\le7\), so \(P(3\le X\le7)=P(X\le7)-P(X\le2)\approx0.7538\). 4. \(P(X>10)=1-P(X\le10)\approx0.0115\).

a) \(0.1800\). b) \(0.7538\). c) \(0.0115\).

- Express the “at least” event using a complement. - Use a difference of cumulative probabilities for the interval. - Check the value immediately below the lower endpoint.

1. \(P(X\le15)\approx0.8369\). 2. \(P(X\ge10)=1-P(X\le9)\approx0.8363\). 3. \(P(8\le X\le14)=P(X\le14)-P(X\le7)\approx0.7028\).

a) \(0.8369\). b) \(0.8363\). c) \(0.7028\).

- Use a complement for the upper-tail probability. - Convert the open interval to the included integer values before using a CDF difference. - Subtract the cumulative probability through the value just below the lower included count.

1. \(P(X\le75)\approx0.1314\). 2. \(P(X>85)=1-P(X\le85)\approx0.0804\). 3. Because \(X\) is integer-valued, \(78<X<88\) means \(79\le X\le87\). 4. Therefore, \(P(78<X<88)=P(X\le87)-P(X\le78)\approx0.6287\).

a) \(0.1314\). b) \(0.0804\). c) \(0.6287\).

- Use one binomial model for all three events. - “Every” means the endpoint count \(X=20\). - Use a complement for “more than \(15\).”

1. Let \(X\sim\operatorname{Bin}(20,0.75)\). 2. \(P(X=20)=(0.75)^{20}\approx0.0032\). 3. \(P(X=15)=\binom{20}{15}(0.75)^{15}(0.25)^5\approx0.2023\). 4. \(P(X>15)=1-P(X\le15)\approx0.4148\).

a) \(0.0032\). b) \(0.2023\). c) \(0.4148\).

- Let success mean a defective component. - Use a point probability for exactly four defects. - Translate “fewer than \(4\)” into the largest included integer count.

1. Let \(X\sim\operatorname{Bin}(80,0.05)\). 2. \(P(X=0)=(0.95)^{80}\approx0.0165\). 3. \(P(X=4)=\binom{80}{4}(0.05)^4(0.95)^{76}\approx0.2004\). 4. Fewer than four means \(X\le3\), so \(P(X<4)\approx0.4284\).

a) \(0.0165\). b) \(0.2004\). c) \(0.4284\).

- Convert the no-show probability to an attendance probability. - Determine the largest attendance count that fits on the bus. - Use a cumulative binomial probability for all acceptable attendance counts.

1. Each customer arrives with probability \(0.90\), so \(X\sim\operatorname{Bin}(54,0.90)\). 2. Everyone has a seat exactly when \(X\le50\). 3. Therefore, \(P(X\le50)\approx0.8015\).

\(P(\text{everyone has a seat})\approx0.8015\).

- Use \(n=150\) and \(p=0.06\). - Use a point probability for exactly nine defects. - Use a complement for the upper tail.

1. \(X\sim\operatorname{Bin}(150,0.06)\). 2. \(P(X=9)=\binom{150}{9}(0.06)^9(0.94)^{141}\approx0.1359\). 3. \(P(X\le5)\approx0.1083\). 4. \(P(X\ge12)=1-P(X\le11)\approx0.1909\).

a) \(0.1359\). b) \(0.1083\). c) \(0.1909\).

- Distinguish “more than” from “at least.” - Which whole-number values satisfy a strict inequality in part b)? - How might you use cumulative probabilities to cover the interval in part c)?

1. \(X\sim\operatorname{Bin}(80,0.25)\). 2. \(P(X>20)=1-P(X\le20)\approx0.4403\). 3. \(P(X<15)=P(X\le14)\approx0.0740\). 4. \(P(18\le X\le22)=P(X\le22)-P(X\le17)\approx0.4811\).

a) \(0.4403\). b) \(0.0740\). c) \(0.4811\).

- Translate each verbal event into integer bounds for \(X\). - Use a complement for the upper tail. - Use a CDF difference for the inclusive interval.

1. \(X\sim\operatorname{Bin}(60,0.80)\). 2. \(P(X=50)=\binom{60}{50}(0.80)^{50}(0.20)^{10}\approx0.1102\). 3. \(P(X<45)=P(X\le44)\approx0.1306\). 4. \(P(X\ge52)=1-P(X\le51)\approx0.1268\). 5. \(P(46\le X\le54)=P(X\le54)-P(X\le45)\approx0.7814\).

a) \(0.1102\). b) \(0.1306\). c) \(0.1268\). d) \(0.7814\).

- Use one binomial model for all four parts. - Use a complement for “more than \(20\).” - Use a cumulative difference for the inclusive interval.

1. \(X\sim\operatorname{Bin}(150,0.12)\). 2. \(P(X=18)\approx0.0998\). 3. \(P(X\le15)\approx0.2715\). 4. \(P(X>20)=1-P(X\le20)\approx0.2587\). 5. \(P(10\le X\le25)=P(X\le25)-P(X\le9)\approx0.9543\).

a) \(0.0998\). b) \(0.2715\). c) \(0.2587\). d) \(0.9543\).

- Count failures rather than successes if that makes the event simpler. - Translate the flower target into a condition on the number of bulbs that fail. - Use the corresponding binomial cumulative probability.

1. Let \(Y\) be the number of bulbs that fail to sprout. Then \(Y\sim\operatorname{Bin}(80,0.15)\). 2. At least \(65\) flowers are produced exactly when at most \(15\) bulbs fail to sprout. 3. Therefore, \(P(Y\le15)\approx0.8625\).

\(P(\text{at least 65 flowers})\approx0.8625\).

- Use \(n=80\) and \(p=0.025\). - “At most \(3\)” is a lower-tail event. - Use the complement of zero ticket holders for part c).

1. \(X\sim\operatorname{Bin}(80,0.025)\). 2. \(P(X=2)=\binom{80}{2}(0.025)^2(0.975)^{78}\approx0.2741\). 3. \(P(X\le3)\approx0.8594\). 4. \(P(X\ge1)=1-P(X=0)=1-(0.975)^{80}\approx0.8681\).

a) \(0.2741\). b) \(0.8594\). c) \(0.8681\).

- Use the supplied one-vehicle occupancy probability. - Translate “more than half of six” into a whole-number demand threshold. - Use a complement for the upper-tail probability.

1. Let \(X\) be the number of vehicles needing a charger at the selected time. Then \(X\sim\operatorname{Bin}(24,1/6)\). 2. More than half of six stations means at least four stations are needed. 3. Therefore, \(P(X\ge4)=1-P(X\le3)\approx0.5845\).

\(P(X\ge4)\approx0.5845\).

- Translate “at least \(80\)” into an upper-tail event. - What does the complementary event look like as an integer inequality? - Keep the endpoint in the complement consistent with the inclusive threshold.

1. \(X\sim\operatorname{Bin}(450,0.15)\). 2. \(P(X\ge80)=1-P(X\le79)\approx0.0590\).

\(P(X\ge80)\approx0.0590\).

- Translate “outside” the inclusive interval into two disjoint tails. - For each boundary, ask whether equality belongs in the interval or outside it. - How could the two tail probabilities be combined?

1. \(X\sim\operatorname{Bin}(600,0.30)\). 2. Being outside \([165,195]\) means \(X\le164\) or \(X\ge196\). 3. The two events are disjoint, so \(P(X\le164\text{ or }X\ge196)=P(X\le164)+P(X\ge196)\approx0.1672\).

\(0.1672\).

- Use the binomial formulas for center and spread. - Translate “within \(5\)” into an inclusive integer interval centered at the mean. - Use a difference of cumulative probabilities.

1. \(\mu=np=60\), and \(\sigma=\sqrt{np(1-p)}=\sqrt{36}=6\). 2. Being within five of the mean means \(55\le X\le65\). 3. Therefore, \(P(55\le X\le65)=P(X\le65)-P(X\le54)\approx0.6407\).

a) \(\mu=60\), \(\sigma=6\). b) \(0.6407\).

- Express the inclusive interval as a difference of cumulative probabilities. - How do you keep the lower endpoint included when subtracting cumulative probabilities? - Evaluate with binomial CDF technology.

1. \(P(80\le X\le110)=P(X\le110)-P(X\le79)\). 2. Therefore, \(P(80\le X\le110)\approx0.9068\).

\(0.9068\).

- Find \(np\) first. - Compare each endpoint with the mean separately. - For an inclusive interval, which cumulative probability must be subtracted?

1. \(\mu=np=80\). 2. The lower endpoint \(72\) is \(8\) below the mean, while \(90\) is \(10\) above it, so the interval is not symmetric about the mean. 3. \(P(72\le X\le90)=P(X\le90)-P(X\le71)\approx0.8251\).

a) \(\mu=80\). b) \(72\) is \(8\) below the mean and \(90\) is \(10\) above it; the interval is not symmetric. c) \(0.8251\).

- Use the binomial model with the supplied parameters. - Express the inclusive interval as a CDF difference. - Subtract through the integer immediately below the lower endpoint.

1. \(X\sim\operatorname{Bin}(150,0.85)\). 2. \(P(122\le X\le132)=P(X\le132)-P(X\le121)\approx0.7875\).

\(0.7875\).

- Use the supplied binomial model for all three events. - Use a complement for the upper tail. - Use a difference of cumulative probabilities for the inclusive interval.

1. \(X\sim\operatorname{Bin}(400,0.02)\). 2. \(P(X\le5)\approx0.1885\). 3. \(P(X>10)=1-P(X\le10)\approx0.1821\). 4. \(P(6\le X\le12)=P(X\le12)-P(X\le5)\approx0.7497\), using unrounded cumulative values.

a) \(0.1885\). b) \(0.1821\). c) \(0.7497\).

- Translate the strict lower-tail boundary carefully. - Use a complement for the upper tail. - Find the binomial mean before translating “within \(5\).”

1. \(X\sim\operatorname{Bin}(100,0.40)\). 2. \(P(X<35)=P(X\le34)\approx0.1303\). 3. \(P(X\ge45)=1-P(X\le44)\approx0.1789\). 4. The mean is \(\mu=np=40\), so being within five of the mean means \(35\le X\le45\). 5. Thus, \(P(35\le X\le45)=P(X\le45)-P(X\le34)\approx0.7386\).

a) \(0.1303\). b) \(0.1789\). c) \(0.7386\).

- Use a lower-tail CDF for part a). - Use a complement for “at least \(25\).” - Use a cumulative difference for the inclusive interval.

1. \(X\sim\operatorname{Bin}(500,0.04)\). 2. \(P(X\le15)\approx0.1513\). 3. \(P(X\ge25)=1-P(X\le24)\approx0.1522\). 4. \(P(18\le X\le22)=P(X\le22)-P(X\le17)\approx0.4314\).

a) \(0.1513\). b) \(0.1522\). c) \(0.4314\).

- The red probability is \(1/4\). - Translate each verbal boundary into an integer inequality. - Use a complement or cumulative difference where efficient.

1. \(X\sim\operatorname{Bin}(40,0.25)\). 2. \(P(X>12)=1-P(X\le12)\approx0.1791\). 3. \(P(8\le X\le12)=P(X\le12)-P(X\le7)\approx0.6389\). 4. \(P(X<5)=P(X\le4)\approx0.0160\).

a) \(0.1791\). b) \(0.6389\). c) \(0.0160\).

- Use the per-holder attendance probability in the binomial model. - How does the number of arrivals compare with the number of available seats when the section is over capacity? - Evaluate the upper tail using a complement.

1. Let \(X\) be the number of ticket holders who attend. Then \(X\sim\operatorname{Bin}(1400,0.85)\). 2. The section is over capacity when \(X>1200\). 3. Therefore, \(P(X>1200)=1-P(X\le1200)\approx0.2169\).

\(P(\text{over capacity})\approx0.2169\).

- Would it be easier to count working drives or defective drives? - How many defective drives could the package contain while still providing enough working drives? - Use the appropriate cumulative binomial probability.

1. Let \(X\) be the number of defective drives. Then \(X\sim\operatorname{Bin}(50,0.10)\). 2. At least \(42\) drives work exactly when at most \(8\) are defective. 3. Therefore, \(P(X\le8)\approx0.9421\).

\(0.9421\).

- Translate the strict inequalities to their included integer counts. - Use a complement for the upper tail. - Use a cumulative difference for the interval in part c).

1. \(X\sim\operatorname{Bin}(200,0.15)\). 2. \(P(X\le25)\approx0.1876\). 3. \(P(X>35)=1-P(X\le35)\approx0.1387\). 4. More than \(25\) but fewer than \(40\) means \(26\le X\le39\), so \(P(26\le X\le39)=P(X\le39)-P(X\le25)\approx0.7788\).

a) \(0.1876\). b) \(0.1387\). c) \(0.7788\).

- For “fewer than \(10\),” which whole-number outcomes qualify? - Use a complement for “at least \(15\).” - Use a cumulative difference for the inclusive interval.

1. \(X\sim\operatorname{Bin}(120,0.12)\). 2. \(P(X<10)=P(X\le9)\approx0.0782\). 3. \(P(X\ge15)=1-P(X\le14)\approx0.4745\). 4. \(P(12\le X\le20)=P(X\le20)-P(X\le11)\approx0.7403\).

a) \(0.0782\). b) \(0.4745\). c) \(0.7403\).

- Match the information in the problem to \(n\), \(p\), and \(k\). - Count the arrangements containing exactly two wins. - Check all four binomial conditions, including whether spins affect one another.

1. The binomial parameters are \(n=6\), \(p=0.30\), and \(k=2\). 2. \(P(X=2)=\binom{6}{2}(0.30)^2(0.70)^4=0.324135\). 3. A binomial model requires a fixed number of trials, two relevant outcomes per trial, independent trials, and a constant success probability. All four conditions are stated here.

a) \(P(X=2)=0.324135\). b) The model has fixed \(n=6\), win or loss on each spin, independent spins, and constant \(p=0.30\).

- Use a point probability for exactly fifteen returns. - What whole-number counts satisfy “fewer than \(10\)”? - Use a complement for “more than \(20\).”

1. Let \(X\sim\operatorname{Bin}(50,0.30)\). 2. \(P(X=15)=\binom{50}{15}(0.30)^{15}(0.70)^{35}\approx0.1223\). 3. \(P(X<10)=P(X\le9)\approx0.0402\). 4. \(P(X>20)=1-P(X\le20)\approx0.0478\).

a) \(0.1223\). b) \(0.0402\). c) \(0.0478\).

- Use a cumulative probability for part a) and a CDF difference for part b). - Find the binomial mean before translating “within \(2\).” - Keep all interval endpoints inclusive.

1. \(X\sim\operatorname{Bin}(200,0.05)\). 2. \(P(X\le12)\approx0.7965\). 3. \(P(5\le X\le15)=P(X\le15)-P(X\le4)\approx0.9292\). 4. The mean is \(\mu=np=10\), so being within two of the mean means \(8\le X\le12\). 5. Thus, \(P(8\le X\le12)=P(X\le12)-P(X\le7)\approx0.5832\).

a) \(0.7965\). b) \(0.9292\). c) \(0.5832\).

- Count the total balls and the green balls in the urn. - How does replacement affect the probability of green on later draws? - Decide how to count sequences containing exactly two green results.

1. The urn contains \(10\) balls, including \(3\) green balls, so \(p=\frac{3}{10}=0.30\). 2. Replacement makes the six draws independent with the same success probability, so \(X\sim\operatorname{Bin}(6,0.30)\). 3. \(P(X=2)=\binom{6}{2}(0.30)^2(0.70)^4=0.324135\). 4. As a percentage rounded to one decimal place, the probability is \(32.4\%\).

Probability of green on one draw: \(\frac{3}{10}\). Probability of exactly two green draws: \(32.4\%\).

- Count the yellow and blue sections in the spinner. - For each part, treat the named color as success and all other colors as failure. - Apply the binomial point-probability formula.

1. From the spinner, \(P(\text{yellow})=\frac{3}{6}=\frac{1}{2}\) and \(P(\text{blue})=\frac{2}{6}=\frac{1}{3}\). 2. \(P(\text{exactly 4 yellow})=\binom{6}{4}\left(\frac{1}{2}\right)^4\left(\frac{1}{2}\right)^2=\frac{15}{64}=23.4375\%\approx23.44\%\). 3. \(P(\text{exactly 2 blue})=\binom{6}{2}\left(\frac{1}{3}\right)^2\left(\frac{2}{3}\right)^4=\frac{80}{243}\approx32.92\%\).

a) \(23.44\%\). b) \(32.92\%\).

- Count the total sections and the red sections. - Identify \(n\), \(p\), and \(k\). - Use the binomial formula for exactly three red outcomes. - Round the final decimal to three places.

1. Four of the \(10\) sections are red, so \(p=\frac{4}{10}=0.4\), and \(1-p=0.6\). 2. Let \(X\) be the number of red outcomes in \(8\) spins. Then \(X\sim\operatorname{Bin}(8,0.4)\). 3. \(P(X=3)=\binom{8}{3}(0.4)^3(0.6)^5=56\cdot(0.4)^3(0.6)^5=0.27869184\approx0.279\).

\(0.279\).

- Count the blue sectors in the spinner. - Which values satisfy “at most one” and “at least three”? - Use the one-spin probability to form the count distribution.

1. Two of the five sectors are blue, so \(p=\frac{2}{5}=0.40\) and \(X\sim\operatorname{Bin}(5,0.40)\). 2. \(P(X\le1)=(0.60)^5+5(0.40)(0.60)^4=0.33696\). 3. \(P(X\ge3)=P(X=3)+P(X=4)+P(X=5)=0.31744\).

a) One-spin blue probability \(\frac{2}{5}=0.40\); \(P(X\le1)=0.33696=33.696\%\). b) \(P(X\ge3)=0.31744=31.744\%\).

- Count the sections of each requested color. - Divide by the total number of sections to find the success probabilities. - Use the binomial formula for each exact number of outcomes. - Use \(n=10\) and the specified value of \(k\) in each part.

1. Three of the eight sections are red, so \(p_R=\frac38=0.375\). One section is yellow, so \(p_Y=\frac18=0.125\). 2. For part a), \(P(X=4)=\binom{10}{4}(0.375)^4(0.625)^6\approx0.247528\), or \(24.75\%\). 3. For part b), \(P(Y=2)=\binom{10}{2}(0.125)^2(0.875)^8\approx0.241600\), or \(24.16\%\).

a) \(24.75\%\) b) \(24.16\%\)

- Count the sections for the color named in each part. - Think about the two outcomes for each spin: the chosen color or another color. - How many different orders can give the required number of spins of that color?

1. From the spinner, \(p_{\text{yellow}}=\frac{4}{8}=0.5\), \(p_{\text{blue}}=\frac{3}{8}=0.375\), and \(p_{\text{red}}=\frac{1}{8}=0.125\). 2. For a), \(P(X=2)=\binom{5}{2}(0.375)^2(0.625)^3\approx0.3433\). 3. For b), \(P(Y=4)=\binom{8}{4}(0.5)^4(0.5)^4=\frac{70}{256}\approx0.2734\). 4. For c), \(P(Z=1)=\binom{4}{1}(0.125)(0.875)^3\approx0.3350\).

a) \(0.3433\). b) \(0.2734\). c) \(0.3350\).

- Read the chance of each named color from the wheel. - Decide which counts satisfy each phrase, “at most” or “at least.” - Can you use a complement to reduce the number of cases to calculate?

1. The spinner has \(5\) blue sections out of \(10\), so the blue probability is \(0.5\). For \(B\sim\operatorname{Bin}(12,0.5)\), \(P(B\le4)=\sum_{k=0}^{4}\binom{12}{k}(0.5)^{12}\approx0.1938\). 2. There are \(2\) red sections out of \(10\), so the red probability is \(0.2\). For \(R\sim\operatorname{Bin}(12,0.2)\), \(P(R\ge2)=1-P(R=0)-P(R=1)=1-(0.8)^{12}-12(0.2)(0.8)^{11}\approx0.7251\).

a) \(0.1938\). b) \(0.7251\).

- Find the chance of blue on a single spin by reading the wheel. - How many spins are there, and how many must be blue? - How many orders can contain exactly that many blue outcomes?

1. One of the five equal sections is blue, so \(p=\frac{1}{5}=0.2\). There are \(6\) spins, with \(3\) required blue results. 2. \(P(X=3)=\binom{6}{3}(0.2)^3(0.8)^3\). 3. Since \(\binom{6}{3}=20\), the probability is \(20\cdot0.008\cdot0.512=0.08192\).

\(P(X=3)=0.08192\), or \(8.192\%\).

- Read the chance of yellow from the wheel. - What does “at most” mean for the possible numbers of yellow results? - Which repeated-trial distribution fits independent spins with an unchanging chance of yellow?

1. There are \(2\) yellow sections out of \(6\), so \(p=\frac{1}{3}\). 2. Let \(X\sim\operatorname{Bin}(50,\frac{1}{3})\). 3. \(P(X\le10)=\sum_{k=0}^{10}\binom{50}{k}\left(\frac{1}{3}\right)^k\left(\frac{2}{3}\right)^{50-k}\approx0.028440\). 4. Rounded to three decimal places, this is \(0.028\).

\(0.028\).

- Treat rolling a six as a success on each roll. - What are the possible numbers of successes in three rolls? - After finding the four probabilities, check whether they account for every possible outcome.

1. \(X\sim\operatorname{Bin}(3,\frac{1}{6})\). 2. \(P(X=0)=\left(\frac{5}{6}\right)^3=\frac{125}{216}\). 3. \(P(X=1)=3\cdot\frac{1}{6}\left(\frac{5}{6}\right)^2=\frac{75}{216}\). 4. \(P(X=2)=3\left(\frac{1}{6}\right)^2\frac{5}{6}=\frac{15}{216}\). 5. \(P(X=3)=\left(\frac{1}{6}\right)^3=\frac{1}{216}\). 6. The probabilities add to \(\frac{125+75+15+1}{216}=1\).

a) \(P(X=0)=\frac{125}{216}\), \(P(X=1)=\frac{75}{216}\), \(P(X=2)=\frac{15}{216}\), and \(P(X=3)=\frac{1}{216}\). b) \(\frac{125+75+15+1}{216}=1\).

- Find the tallest bar to identify the most likely count. - Which counts satisfy the inequality in part b)? - Add their bar heights rather than calculating a new binomial formula.

1. The tallest bar is at \(X=2\), with probability \(\frac{6}{16}\). 2. For at least three heads, add the bars at \(3\) and \(4\): \(P(X\ge3)=\frac{4}{16}+\frac{1}{16}=\frac{5}{16}\).

a) \(X=2\). b) \(P(X\ge3)=\frac{5}{16}=0.3125\).

- Write all possible three-toss sequences. - Keep only sequences with exactly the required number of heads. - Compare the number of favorable sequences with the total.

1. The equally likely outcomes with exactly one head are \(HTT\), \(THT\), and \(TTH\). 2. There are \(2^3=8\) equally likely outcomes in all, so \(P(X=1)=\frac{3}{8}\).

Outcomes: \(HTT\), \(THT\), \(TTH\). Probability: \(\frac{3}{8}\).

- List the integer counts in each event. - Identify which count appears in one event but not the other. - Find the probability of that boundary count.

1. The event \(X\ge2\) includes exactly \(2\), \(3\), or \(4\) correct answers; \(X>2\) includes only \(3\) or \(4\). 2. The difference in probabilities is exactly \(P(X=2)\). 3. \(P(X=2)=\binom{4}{2}(0.5)^4=\frac{6}{16}=0.375\).

The difference is \(0.375\). The first event includes exactly two correct answers; the second does not.

- Compare what remains in the bag after the first draw. - Ask whether the chance of red on the second draw stays the same. - What does replacement change about the trials?

1. Without replacement, the bag's contents change after the first draw, so the chance of red on the second draw depends on what happened first. 2. Thus, the two draws do not satisfy the independent, constant-probability conditions for a binomial count. 3. With replacement, each draw would have red probability \(\frac{2}{5}\) independently, giving a binomial count with \(n=2\) and \(p=\frac{2}{5}\).

Without replacement, no. With replacement, yes: \(X\sim\operatorname{Bin}\left(2,\frac{2}{5}\right)\).

- First find the probability that all \(12\) glasses in one box are bubble-free. - Use that box-level probability as the success probability for the second binomial model. - “At least \(5\)” includes \(5\) and \(6\).

1. One glass is bubble-free with probability \(0.97\). 2. Thus, the probability one box is entirely bubble-free is \(p_B=0.97^{12}\approx0.6938.\) 3. Let \(Y\sim\operatorname{Bin}(6,p_B)\). 4. Then \(P(Y\ge5)=\binom65p_B^5(1-p_B)+p_B^6\approx0.4070.\)

a) \(0.97^{12}\approx0.6938\). b) \(\binom65(0.97^{12})^5(1-0.97^{12})+(0.97^{12})^6\approx0.4070\).

- Model each player’s shots with a binomial distribution. - Identify \(n\), \(p\), and the relevant values of the random variable for each player. - For Player A, “at least \(3\)” includes more than one case. - Use the binomial probability formula for each required case.

1. For Player A, let \(X\) be the number of made shots. Then \(X\) has a binomial distribution with \(n=4\) and \(p=0.90\). 2. \(P(X\ge 3)=P(X=3)+P(X=4)\). 3. \(P(X=3)=\binom{4}{3}(0.90)^3(0.10)=0.2916\), and \(P(X=4)=(0.90)^4=0.6561\). 4. Therefore, \(P(X\ge 3)=0.2916+0.6561=0.9477\). 5. For Player B, let \(Y\) be the number of made shots. Then \(Y\) has a binomial distribution with \(n=6\) and \(p=0.60\). 6. \(P(Y=4)=\binom{6}{4}(0.60)^4(0.40)^2=0.31104\). 7. Since \(0.9477>0.31104\), Event 1 is more likely.

Event 1 is more likely. Its probability is \(94.77\%\), compared with approximately \(31.10\%\) for Event 2.

- Determine the success probability for one question in each strategy. - Identify the number of questions in each strategy. - “At least \(3\)” includes \(3\), \(4\), and \(5\). - Calculate the required binomial probabilities and compare them.

1. For Strategy A, let \(X\sim\operatorname{Bin}(5,0.25)\). Then \(P(X\ge 3)=\sum_{k=3}^{5}\binom{5}{k}(0.25)^k(0.75)^{5-k}=\frac{106}{1024}\approx 0.103516\). 2. For Strategy B, let \(Y\sim\operatorname{Bin}(10,0.50)\). Then \(P(Y=5)=\binom{10}{5}(0.50)^{10}=\frac{252}{1024}\approx 0.246094\). 3. Since \(0.246094>0.103516\), Strategy B has the greater success probability.

Strategy B has the greater success probability: approximately \(24.61\%\), compared with approximately \(10.35\%\) for Strategy A.

- Use the binomial model with \(n=20\) and \(p=0.05\). - “At most \(2\)” includes \(0,1,2\). - Use the complement of no defective components for “at least \(1\).”

1. \(X\sim\operatorname{Bin}(20,0.05)\). 2. \(P(X=1)=\binom{20}{1}(0.05)(0.95)^{19}\approx0.3774.\) 3. \(P(X\le2)=\sum_{k=0}^{2}\binom{20}{k}(0.05)^k(0.95)^{20-k}\approx0.9245.\) 4. \(P(X\ge1)=1-(0.95)^{20}\approx0.6415.\)

a) \(\binom{20}{1}(0.05)(0.95)^{19}\approx0.3774\). b) \(\sum_{k=0}^{2}\binom{20}{k}(0.05)^k(0.95)^{20-k}\approx0.9245\). c) \(1-(0.95)^{20}\approx0.6415\).

- Translate each verbal condition into a set of integer hit counts. - Use a binomial tail or finite sum. - Keep the endpoint inequalities strict or inclusive as stated.

1. \(X\sim\operatorname{Bin}(10,0.7)\). 2. \(P(X\ge8)=\sum_{k=8}^{10}\binom{10}{k}(0.7)^k(0.3)^{10-k}\approx0.3828.\) 3. The event \(5<X<9\) is \(X\in\{6,7,8\}\), so \(P(5<X<9)=\sum_{k=6}^{8}\binom{10}{k}(0.7)^k(0.3)^{10-k}\approx0.7004.\)

a) \(\sum_{k=8}^{10}\binom{10}{k}(0.7)^k(0.3)^{10-k}\approx0.3828\). b) \(\sum_{k=6}^{8}\binom{10}{k}(0.7)^k(0.3)^{10-k}\approx0.7004\).

- Model the number of red results first. - Use a complement for “at least \(2\).” - Translate “at least \(9\) nonred” into a condition on the red count.

1. For red, \(X\sim\operatorname{Bin}(10,0.25)\). 2. \(P(X=2)=\binom{10}{2}(0.25)^2(0.75)^8\approx0.2816.\) 3. \(P(X\le1)=(0.75)^{10}+10(0.25)(0.75)^9\approx0.2440.\) 4. \(P(X\ge2)=1-P(X\le1)\approx0.7560.\) 5. At least \(9\) nonred results means at most \(1\) red result, so the probability is the same as in part b), about \(0.2440\).

a) \(\binom{10}{2}(0.25)^2(0.75)^8\approx0.2816\). b) \((0.75)^{10}+10(0.25)(0.75)^9\approx0.2440\). c) \(1-[(0.75)^{10}+10(0.25)(0.75)^9]\approx0.7560\). d) \((0.75)^{10}+10(0.25)(0.75)^9\approx0.2440\).

- Count sixes as successes. - Use complements for the cumulative events. - Translate the number of non-sixes into the corresponding number of sixes.

1. Let \(X\) be the number of sixes. Then \(X\sim\operatorname{Bin}(6,\frac{1}{6})\). 2. \(P(X=1)=6\left(\frac{1}{6}\right)\left(\frac{5}{6}\right)^5=\left(\frac{5}{6}\right)^5\approx0.4019\). 3. \(P(X=0)=\left(\frac{5}{6}\right)^6\approx0.3349\). 4. \(P(X\ge2)=1-P(X=0)-P(X=1)\approx0.2632\). 5. “A number other than \(6\) at most \(5\) times” is equivalent to at least one \(6\), so \(P(X\ge1)=1-\left(\frac{5}{6}\right)^6\approx0.6651\).

a) \(\left(\frac{5}{6}\right)^5\approx0.4019\). b) \(\left(\frac{5}{6}\right)^6\approx0.3349\). c) \(1-\left(\frac{5}{6}\right)^6-\left(\frac{5}{6}\right)^5\approx0.2632\). d) \(1-\left(\frac{5}{6}\right)^6\approx0.6651\).

- Recall the four conditions for a binomial setting. - Use a complement or binomial upper tail for part b). - When using the interval around the mean, remember that \(X\) can take only integer values.

1. A binomial model requires a fixed number of trials, two relevant outcomes per trial, independent trials, and a constant success probability. Here, \(n=40\) and \(p=0.75\). 2. \(P(X\ge35)=0.04327398\ldots\approx0.0433\). 3. The mean is \(\mu=np=30\), and the standard deviation is \(\sigma=\sqrt{np(1-p)}=\sqrt{7.5}\approx2.7386\). 4. The interval is approximately \([27.2614,32.7386]\). Since \(X\) is an integer count, this corresponds to \(28\le X\le32\). 5. Therefore, \(P(28\le X\le32)=0.63891160\ldots\approx0.6389\).

a) Fixed \(n\), two relevant outcomes, independence, and constant \(p\); \(n=40\), \(p=0.75\). b) \(P(X\ge35)\approx0.0433\). c) \(\mu=30\) and \(\sigma=\sqrt{7.5}\approx2.7386\). d) \(P(28\le X\le32)\approx0.6389\).

- For part a), compare a direct sum with using the complement of \(0\) or \(1\) success. - What must happen on the first \(3\) spins for the first success to occur on spin \(4\)? - Compare the expression in part c) with the binomial probability formula.

1. The success probability is \(p=\frac{1}{5}=0.2\). 2. For part a), let \(X\) be the number of successes in \(6\) spins. Then \(P(X\ge 2)=1-[P(X=0)+P(X=1)]\). 3. \(P(X\ge 2)=1-[(0.8)^6+6(0.2)(0.8)^5]=0.34464\). 4. For the first success to occur on the fourth spin, the first \(3\) spins must be failures and the fourth must be a success. The last \(2\) spins can have any outcomes. 5. Therefore, the probability is \((0.8)^3(0.2)=0.1024\). 6. In part c), \(n=6\), \(k=3\), and \(p=0.2\). Event \(A\) is getting exactly \(3\) blue results in \(6\) spins.

a) \(P(X\ge 2)=0.34464\), or approximately \(34.46\%\) b) The probability is \(0.1024\), or \(10.24\%\). c) \(n=6\), \(k=3\), and \(p=0.2\). Event \(A\) is getting exactly \(3\) blue results in \(6\) spins.

- Compare each expression with the general binomial probability formula. - Check the relationship between the upper number in the binomial coefficient and the exponents. - Verify that the success and failure probabilities add to \(1\). - For a valid model, describe repeated independent trials with two relevant outcomes.

1. In a), the coefficient and exponents match \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\). Here \(n=10\), \(k=3\), and \(p=0.15\). For example, a process with a \(15\%\) success probability is repeated \(10\) independent times, and the event is exactly \(3\) successes. 2. In b), the binomial coefficient indicates \(n=12\), but the exponents add to \(4+6=10\), not \(12\). Therefore, the expression is not a valid binomial probability formula. 3. In c), the expression is valid with \(n=10\), \(k=8\), and \(p=0.80\). The factors are written in the opposite order, but multiplication is commutative. For example, a basketball player with an \(80\%\) free-throw success rate takes \(10\) independent shots, and the event is exactly \(8\) made shots.

a) Yes: \(n=10\), \(p=0.15\), \(k=3\). One example is exactly \(3\) successes in \(10\) independent trials with success probability \(0.15\). b) No. The exponents add to \(10\), but the binomial coefficient indicates \(n=12\). c) Yes: \(n=10\), \(p=0.80\), \(k=8\). One example is exactly \(8\) made free throws in \(10\) independent attempts with success probability \(0.80\).

- Check the conditions required for a binomial model. - What values of \(X\) are included in “at most \(1\)”? - Recall the values of \(\binom{n}{0}\) and \(\binom{n}{1}\). - Substitute \(k=0\) and \(k=1\) separately into the binomial probability formula.

1. For part a), there are \(15\) independent trials, each with two outcomes and a constant misshapen probability of \(0.04\). 2. The factor \(\binom{15}{2}\) counts the ways to choose the \(2\) misshapen chocolates, \((0.04)^2\) is the probability for those \(2\) outcomes, and \((0.96)^{13}\) is the probability that the other \(13\) chocolates are not misshapen. Therefore, the expression is correct. 3. For part b), “at most \(1\)” means \(X=0\) or \(X=1\). 4. \(P(X=0)=\binom{15}{0}(0.04)^0(0.96)^{15}=(0.96)^{15}\). 5. \(P(X=1)=\binom{15}{1}(0.04)(0.96)^{14}=15(0.04)(0.96)^{14}\). 6. Their sum is exactly the employee’s expression, so the claim is correct.

a) The expression correctly applies the binomial probability formula with \(n=15\), \(k=2\), and \(p=0.04\). b) The claim is correct because the expression is \(P(X=0)+P(X=1)=P(X\le 1)\).

- Use the binomial point formula for part a). - Use cumulative probabilities for the tails and interval. - Check the endpoint inclusion carefully.

1. \(P(X=6)=\binom{20}{6}(0.3)^6(0.7)^{14}\approx0.1916.\) 2. \(P(X\le4)=\sum_{k=0}^{4}\binom{20}{k}(0.3)^k(0.7)^{20-k}\approx0.2375.\) 3. \(P(X>7)=1-P(X\le7)\approx0.2277.\) 4. \(P(5\le X\le9)=P(X\le9)-P(X\le4)\approx0.7145.\)

a) \(0.1916\). b) \(0.2375\). c) \(0.2277\). d) \(0.7145\).

- First identify the probabilities of a correct and an incorrect answer. - Use the exponents to determine how many questions are represented. - What event is the complement of getting no correct answers? - In part d), identify the two probabilities being subtracted from \(1\).

1. Let \(X\) be the number of correct answers. Then \(X\) has a binomial distribution with \(n=6\) and \(p=\frac{1}{5}\). 2. The expression in part a) is \(P(X=0)\), the probability of answering no questions correctly. 3. The expression in part b) is \(1-P(X=0)=P(X\ge1)\), the probability of answering at least one question correctly. 4. The expression in part c) is \(\binom{6}{1}\frac{1}{5}\left(\frac{4}{5}\right)^5=P(X=1)\), the probability of answering exactly one question correctly. 5. The expression in part d) is \(1-[P(X=0)+P(X=1)]=P(X\ge2)\), the probability of answering at least two questions correctly.

a) Mila answers none of the \(6\) questions correctly. b) Mila answers at least \(1\) question correctly. c) Mila answers exactly \(1\) of the \(6\) questions correctly. d) Mila answers at least \(2\) questions correctly.

- Choose the appropriate success definition for each part. - Use a complement for “at least one.” - “More than \(13\)” means \(14\) or \(15\).

1. For a specified day, the success probability is \(1/7\). 2. \(P(X=2)=\binom{15}{2}\left(\frac17\right)^2\left(\frac67\right)^{13}\approx0.2889.\) 3. \(P(X\ge1)=1-\left(\frac67\right)^{15}\approx0.9010.\) 4. For Monday through Friday, \(p=5/7\), so \(P(X>13)=P(X=14)+P(X=15)\approx0.0450.\)

a) \(0.2889\). b) \(0.9010\). c) \(0.0450\).

- Use combinations to count bit positions in part a). - For part b), add the zero-error and one-error probabilities. - State all four binomial conditions in context.

1. The number of blocks with exactly two ones is \(\binom{20}{2}=190\). 2. If \(X\) is the number of bit errors, then \(X\sim\operatorname{Bin}(20,0.01)\). 3. \(P(X\le1)=(0.99)^{20}+20(0.01)(0.99)^{19}\approx0.9831.\) 4. A binomial model requires a fixed number of bits, two relevant outcomes per bit, a constant error probability, and independent error outcomes.

a) \(190\). b) \((0.99)^{20}+20(0.01)(0.99)^{19}\approx0.9831\). c) Fixed \(n\), two relevant outcomes, constant \(p\), and independence.

- Count one-letter and two-letter prefixes separately. - Count the special prefixes beginning with A. - Use the resulting probability in the binomial formula.

1. There are \((26+26^2)(99)=69{,}498\) possible codes. 2. There are \((1+26)(99)=2673\) special codes, so \(p=\frac{2673}{69498}=\frac1{26}.\) 3. If \(X\) counts special codes, then \(X\sim\operatorname{Bin}(20,1/26)\), so \(P(X=3)=\binom{20}{3}\left(\frac1{26}\right)^3\left(\frac{25}{26}\right)^{17}\approx0.0333.\)

1. \(69{,}498\). 2. \(p=\frac1{26}\). 3. \(\binom{20}{3}(1/26)^3(25/26)^{17}\approx0.0333\).

- Count all possible numeric suffixes and those beginning with \(9\). - Model the ten generated codes with a binomial random variable. - Use a complement for “more than \(2\).”

1. There are \(5\cdot1000=5000\) possible codes and \(5\cdot100=500\) priority codes, so \(p=\frac{500}{5000}=0.1\). 2. If \(Y\sim\operatorname{Bin}(10,0.1)\), then \(P(Y>2)=1-\sum_{k=0}^{2}\binom{10}{k}(0.1)^{k}(0.9)^{10-k}\approx0.0702\).

1. \(p=0.1\). 2. \(P(Y>2)\approx0.0702\).

- Treat Ava and Ben as one block when counting favorable seatings. - Use the one-night probability as the success probability for ten nights. - The probability in part b) is an exact finite decimal.

1. Put five distinct name tiles in a bag and draw all five without replacement; the draw order represents the seating order. 2. There are \(5!=120\) seating orders. Treat Ava and Ben as a block: \(2(4!)=48\) orders place them together. 3. Thus, \(p=48/120=0.4\). 4. Across \(10\) independent nights, let \(X\sim\operatorname{Bin}(10,0.4)\). 5. \(P(X=4)=\binom{10}{4}(0.4)^4(0.6)^6=0.250822656.\)

a) Use five name tiles drawn without replacement; \(p=0.4\). b) \(P(X=4)=0.250822656\).

- Use the binomial model with \(n=25\). - Use a complement for at least two returns. - Expected count is \(np\).

1. For part a), \(X\sim\operatorname{Bin}(25,0.15)\). 2. \(P(X=3)=\binom{25}{3}(0.15)^3(0.85)^{22}\approx0.2174.\) 3. \(P(X=0)=(0.85)^{25}\approx0.0172.\) 4. \(P(X\ge2)=1-P(X=0)-P(X=1)\approx0.9069.\) 5. At \(p=0.15\), \(E(X)=25(0.15)=3.75\). At \(p=0.20\), \(E(X)=25(0.20)=5\), an increase of \(1.25\).

a) (1) \(0.2174\); (2) \(0.0172\); (3) \(0.9069\). b) The expected count increases from \(3.75\) to \(5\), an increase of \(1.25\).

- Count the possible positions of the three correct answers. - Use the binomial coefficient when any three positions may be correct. - Do not use a binomial coefficient for one specified order.

1. The number of patterns is \(\binom{10}{3}=120\). 2. \(P(X=3)=\binom{10}{3}(0.2)^3(0.8)^7=0.201326592\). 3. For the single specified order, \((0.2)^3(0.8)^7=0.0016777216\).

a) \(120\). b) \(0.201326592\). c) \(0.0016777216\).

- Use a combination to count the positions of the four special-handling packages. - Use the same combination inside the exact binomial probability. - Translate “at most” into possible counts before writing the sum.

1. The number of sequences is \(\binom{20}{4}=4845\). 2. \(P(X=4)=\binom{20}{4}(0.15)^4(0.85)^{16}\approx0.1821\). 3. \(P(X\le2)=\sum_{k=0}^{2}\binom{20}{k}(0.15)^k(0.85)^{20-k}\approx0.4049\).

a) \(4845\). b) \(\binom{20}{4}(0.15)^4(0.85)^{16}\approx0.1821\). c) \(\sum_{k=0}^{2}\binom{20}{k}(0.15)^k(0.85)^{20-k}\approx0.4049\).

- Compare the number of rare sixes in the two fixed sequences. - A fixed sequence does not use a binomial coefficient. - An exact success count includes every arrangement with that count.

1. Sequence B is more likely because it contains only three occurrences of the less likely result \(6\). 2. \(P(A)=(1/6)^6=1/46656\). 3. \(P(B)=(1/6)^3(5/6)^3=125/46656\). 4. Exactly three sixes can occur in \(\binom63=20\) positions, so \(P(X=3)=\binom63(1/6)^3(5/6)^3=2500/46656\). 5. Sequence B is only one of the \(20\) ordered patterns included in \(X=3\).

a) Sequence B. b) \(P(A)=1/46656\), \(P(B)=125/46656\). c) \(P(X=3)=2500/46656\); it includes \(20\) possible arrangements, not just Sequence B.

- Compare the frequency of the rare outcome \(D\). - A specified record fixes the defect position. - An exact defect count allows every possible defect position.

1. Record (3) is least likely because every position contains the rare \(D\) outcome. 2. \(P(1)=(0.95)^8\), while \(P(2)=0.05(0.95)^7\). 3. Exactly one defective chip can occur in any of eight positions, so \(P(X=1)=8(0.05)(0.95)^7\). 4. Therefore, the claim is false: the exact-one event has eight times the probability of the one specified record.

a) Record (3). b) \(P(1)=(0.95)^8\), \(P(2)=0.05(0.95)^7\). c) The claim is false; \(P(X=1)=8(0.05)(0.95)^7\).

- Decide whether the positions of the two nongerminating seeds are specified. - Use a binomial coefficient only when any arrangement with the given count is allowed. - Compare one fixed sequence with all sequences having the same count.

1. For part a), any \(13\) of the \(15\) positions may be the germinating seeds, so \(P(X=13)=\binom{15}{13}(0.85)^{13}(0.15)^2.\) 2. Part b specifies one particular ordered pattern, so \(P=(0.85)^{13}(0.15)^2.\) 3. The binomial exact-count probability is \(\binom{15}{13}\) times the probability of that one specified pattern.

a) \(\binom{15}{13}(0.85)^{13}(0.15)^2\). b) \((0.85)^{13}(0.15)^2\).

- State all four binomial conditions. - Identify \(n\), \(p\), and \(k\). - Use the binomial point-probability formula.

1. The model has a fixed \(12\) trials, two relevant outcomes on each shot, constant make probability \(0.75\), and independent shot outcomes. 2. Thus, \(X\sim\operatorname{Bin}(12,0.75)\). 3. \(P(X=9)=\binom{12}{9}(0.75)^9(0.25)^3.\)

Fixed \(n=12\), two relevant outcomes, constant \(p=0.75\), and independence. The probability is \(\binom{12}{9}(0.75)^9(0.25)^3\).

- How does replacement affect the success probability from one trial to the next? - What condition makes repeated trials independent? - For the binomial case, identify \(n\), \(k\), \(p\), and \(1-p\).

1. With replacement, each selection has constant defect probability \(p=\frac{6}{40}=0.15\), and the selections are independent. This is a binomial experiment with \(n=5\). 2. Without replacement, the composition of the container changes, so the defect probability changes and the selections are dependent. This is not a binomial experiment. 3. For the with-replacement case, \(P(X=1)=\binom{5}{1}(0.15)^1(0.85)^4\approx 0.3915\).

a) Binomial with \(n=5\) and \(p=0.15\). The probability of exactly one defective component is \(\binom{5}{1}(0.15)(0.85)^4\approx 0.3915\), or about \(39.15\%\). b) Not binomial because the probability changes without replacement.

- Distinguish a fixed ordered pattern from a count event. - Multiply success and failure probabilities for a fixed sequence. - Use a binomial coefficient only when the failure position is not specified.

1. Part a specifies one sequence with five successes and one failure: \((0.85)^5(0.15).\) 2. Part b is \((0.85)^6.\) 3. Part c specifies four successes followed by two failures: \((0.85)^4(0.15)^2.\) 4. Part d allows the one failure in any of six positions: \(\binom65(0.85)^5(0.15).\)

a) \((0.85)^5(0.15)\). b) \((0.85)^6\). c) \((0.85)^4(0.15)^2\). d) \(\binom65(0.85)^5(0.15)\).

- Translate “more than \(6\)” into the correct binomial tail. - Find the binomial mean and standard deviation before forming the interval. - Use the decimal interval endpoints to select only attainable integer counts.

1. \(Y\sim\operatorname{Bin}(80,0.05)\). 2. \(P(Y>6)=0.10528525\ldots\approx0.1053\). 3. The mean is \(\mu=4\), and the standard deviation is \(\sigma=\sqrt{3.8}\approx1.9494\). 4. The interval \([\mu-\sigma,\mu+\sigma]\) is approximately \([2.0506,5.9494]\), so the integer values are \(3,4,5\). 5. Therefore, \(P(3\le Y\le5)=0.55860417\ldots\approx0.5586\).

1. \(P(Y>6)\approx0.1053\). 2. The integer values are \(3,4,5\), and \(P(3\le Y\le5)\approx0.5586\).

- Read the lower and upper summation limits as possible values of \(Y\). - A leading \(1-\) indicates a complement. - Translate the expression before evaluating it numerically.

1. Part a is \(P(Y\le10)\), the probability of at most \(10\) defective components, and it is approximately \(0.9885\). 2. Part b is \(P(Y\ge3)\), the probability of at least \(3\) defective components, and it is approximately \(0.8817\). 3. Part c is \(P(4\le Y\le7)\), the probability of from \(4\) through \(7\) defective components, and it is approximately \(0.6142\).

a) At most \(10\) defective components; \(0.9885\). b) At least \(3\) defective components; \(0.8817\). c) From \(4\) through \(7\) defective components; \(0.6142\).

- Interpret a cumulative probability as a sum of point probabilities. - Split the event \(X\le b\) into two nonoverlapping parts. - Which values included in \(F(b)\) are not included in \(F(a)\)? - Subtract the common portion.

1. By definition, \(F(b)=P(X\le b)\) and \(F(a)=P(X\le a)\). 2. The event \(\{X\le b\}\) is the disjoint union of \(\{X\le a\}\) and \(\{a<X\le b\}\). 3. Therefore, additivity for disjoint events gives \(P(X\le b)=P(X\le a)+P(a<X\le b)\). 4. Subtracting \(P(X\le a)\) from both sides yields \(P(a<X\le b)=P(X\le b)-P(X\le a)=F(b)-F(a)\).

Because \(\{X\le b\}\) is the disjoint union of \(\{X\le a\}\) and \(\{a<X\le b\}\), \(F(b)=F(a)+P(a<X\le b)\). Therefore, \(P(a<X\le b)=F(b)-F(a)\).

- A binomial count can take only whole-number values. - How does a strict inequality change which endpoint belongs in the sum? - Decide the first and last allowed values before writing the summation limits. - Each term of the sum is a binomial probability for one possible count.

1. Because \(X\) is integer-valued, \(P(X<85)=P(X\le 84)\). 2. \(P(X<85)=\sum_{i=0}^{84}\binom{150}{i}(0.6)^i(0.4)^{150-i}\approx 0.1794\). 3. \(P(X\ge 100)=\sum_{i=100}^{150}\binom{150}{i}(0.6)^i(0.4)^{150-i}\approx 0.0555\). 4. For part c), the included values are \(80\) through \(109\). 5. \(P(80\le X<110)=\sum_{i=80}^{109}\binom{150}{i}(0.6)^i(0.4)^{150-i}\approx 0.9587\).

a) \(\sum_{i=0}^{84}\binom{150}{i}(0.6)^i(0.4)^{150-i}\approx 0.1794\) b) \(\sum_{i=100}^{150}\binom{150}{i}(0.6)^i(0.4)^{150-i}\approx 0.0555\) c) \(\sum_{i=80}^{109}\binom{150}{i}(0.6)^i(0.4)^{150-i}\approx 0.9587\)

- Rewrite each absolute-value condition as an interval. - Intersect the interval with the integer support of \(X\). - Use a complement for part b)(2).

1. \(|X-20|\le3\) gives \(17\le X\le23\), so the possible values are \(17\) through \(23\). 2. \(P(17\le X\le23)\approx0.6340\). 3. The complement is \(P(|X-20|>3)\approx0.3660\). 4. \(|X-18|<2\) gives \(16<X<20\), so \(X\in\{17,18,19\}\), with probability approximately \(0.2731\).

a) \(\{17,18,19,20,21,22,23\}\). b) (1) \(0.6340\); (2) \(0.3660\); (3) \(0.2731\).

- Use \(n=8\) and \(p=0.70\). - “At least \(7\)” includes \(7\) and \(8\). - Part c is the complement of at least \(6\).

1. Let \(X\sim\operatorname{Bin}(8,0.70)\). 2. \(P(X=6)\approx0.2965\). 3. \(P(X\ge7)\approx0.2553\). 4. \(P(X\le5)\approx0.4482\).

a) \(0.2965\). b) \(0.2553\). c) \(0.4482\).

- Read the summation limits as allowed success counts. - Add the corresponding binomial point probabilities. - Preserve the exact fraction where requested.

1. Part a represents \(3\), \(4\), or \(5\) successes: \(P(3\le X\le5)\approx0.3172.\) 2. Part b represents \(4\), \(5\), or \(6\) successes: \(P(4\le Y\le6)=\frac{2211}{4096}.\)

a) \(3\), \(4\), or \(5\) successes; \(0.3172\). b) \(4\), \(5\), or \(6\) successes; \(\frac{2211}{4096}\).

- First determine the one-spin success probability for each event. - Use a binomial model when only the number of successes matters. - Do not use a binomial coefficient for one specified order.

1. Four of the eight labels are prime, so \(p=1/2\). Thus, \(P(X=4)=\binom{10}{4}(1/2)^{10}\approx0.2051.\) 2. \(P(X\le3)=\sum_{k=0}^{3}\binom{10}{k}(1/2)^{10}\approx0.1719.\) 3. Two labels exceed \(6\), so \(p=1/4\). Therefore, \(P(Y\ge8)=\sum_{k=8}^{10}\binom{10}{k}(1/4)^k(3/4)^{10-k}\approx0.0004.\) 4. The order is fixed, so \((1/2)^3(1/2)^7=(1/2)^{10}\approx0.0010.\)

1. \(0.2051\). 2. \(0.1719\). 3. \(0.0004\). 4. \(0.0010\).

- Use the exponents and any binomial coefficient to identify the number of draws. - What does the absence of a binomial coefficient imply about order? - Interpret a summation as a cumulative binomial probability. - For an expression beginning with \(1-\), identify the complementary event.

1. The first expression has \(n=15\) and \(p=0.2\). It represents drawing exactly \(4\) red marbles. 2. The second expression has \(n=12\) and \(p=0.4\). It represents drawing at most \(2\) red marbles. 3. The exponents in the third expression total \(10\), so \(n=10\) and \(p=0.3\). Because there is no binomial coefficient, the expression represents one specified sequence with \(3\) red draws and \(7\) non-red draws, such as red on the first \(3\) draws and non-red on the last \(7\). 4. The fourth expression has \(n=20\) and \(p=0.9\). The sum is \(P(X\le 13)\), so the full expression is \(P(X\ge 14)\), representing at least \(14\) red draws.

1. \(n=15\), \(p=0.2\); \(E\): exactly \(4\) red marbles are drawn. 2. \(n=12\), \(p=0.4\); \(E\): at most \(2\) red marbles are drawn. 3. \(n=10\), \(p=0.3\); \(E\): one specified sequence of \(3\) red draws and \(7\) non-red draws. 4. \(n=20\), \(p=0.9\); \(E\): at least \(14\) red marbles are drawn.

- Find the probability of one correct answer. - Relate the mode of a binomial distribution to \((n+1)p\). - Check the two endpoint probabilities, \(P(X=0)\) and \(P(X=n)\). - Compare all incorrect answers with all correct answers.

1. The number of correct answers is binomial with \(n=20\) and \(p=\frac{1}{5}=0.2\). 2. Since \((n+1)p=21(0.2)=4.2\) is not an integer, the unique mode is \(\lfloor 4.2\rfloor=4\). 3. Binomial probabilities increase up to their mode and then decrease, so the least likely count occurs at an endpoint. Compare \(P(X=0)=(0.8)^{20}\approx0.0115\) and \(P(X=20)=(0.2)^{20}\approx1.05\times10^{-14}\). 4. Because \((0.2)^{20}<(0.8)^{20}\), \(k=20\) is least likely.

The most likely number is \(k=4\). The least likely number is \(k=20\).

- Use a binomial model with \(n=20\) and \(p=0.85\). - Translate the verbal boundaries into integer counts. - Use cumulative probabilities or complements for the tails.

1. \(X\sim\operatorname{Bin}(20,0.85)\). 2. \(P(X=18)\approx0.2293\). 3. \(P(X\le16)\approx0.3523\). 4. \(P(X\ge15)\approx0.9327\). 5. \(P(17\le X\le19)\approx0.6090\).

a) \(0.2293\). b) \(0.3523\). c) \(0.9327\). d) \(0.6090\).

- “At least \(2\)” is easier through a complement. - Include both zero and one defect in the complementary event. - Keep the exact expression until the final rounding.

1. \(X\sim\operatorname{Bin}(15,0.04)\). 2. Use the complement of zero or one defect: \(P(X\ge2)=1-(0.96)^{15}-15(0.04)(0.96)^{14}.\) 3. Therefore, \(P(X\ge2)\approx0.1191.\)

\(1-(0.96)^{15}-15(0.04)(0.96)^{14}\approx0.1191\).

- Translate “more than \(3\)” into an integer tail. - Could calculating the event that avoids rejection be easier? - Convert the final probability to a percentage only after calculating it.

1. Let \(X\sim\operatorname{Bin}(20,0.10)\). 2. A package is rejected when \(X\ge4\), so \(P(X\ge4)=1-\sum_{k=0}^{3}\binom{20}{k}(0.10)^k(0.90)^{20-k}\approx0.1330\). 3. As a percentage, this is approximately \(13.3\%\).

\(P(X\ge4)=1-\sum_{k=0}^{3}\binom{20}{k}(0.10)^k(0.90)^{20-k}\approx0.1330\), so approximately \(13.3\%\) of packages are rejected.

- Translate the word “fewer” carefully. - Use the binomial cumulative probability through \(7\). - Round only the final result.

1. Let \(X\sim\operatorname{Bin}(12,0.75)\). 2. “Fewer than \(8\)” means \(X\le7\). 3. \(P(X\le7)=\sum_{k=0}^{7}\binom{12}{k}(0.75)^k(0.25)^{12-k}\approx0.1576.\)

\(P(X\le7)\approx0.1576\).

- Simplify the two probability factors when \(p=1-p=0.5\). - Recall the symmetry of binomial coefficients. - Pair the upper-tail probabilities with corresponding lower-tail probabilities.

1. When \(p=0.5\), \(P(X=k)=\binom{n}{k}(0.5)^k(0.5)^{n-k}=\binom{n}{k}(0.5)^n\). 2. Similarly, \(P(X=n-k)=\binom{n}{n-k}(0.5)^{n-k}(0.5)^k=\binom{n}{n-k}(0.5)^n\). 3. Since \(\binom{n}{k}=\binom{n}{n-k}\), it follows that \(P(X=k)=P(X=n-k)\). 4. For \(n=6\), \(P(X\ge4)=\left(\binom{6}{4}+\binom{6}{5}+\binom{6}{6}\right)(0.5)^6=\frac{22}{64}=0.34375\). 5. The symmetry gives \(P(X=4)=P(X=2)\), \(P(X=5)=P(X=1)\), and \(P(X=6)=P(X=0)\). Therefore, \(P(X\ge4)=P(X\le2)\).

a) Because \(\binom{n}{k}=\binom{n}{n-k}\) and both probabilities contain the factor \((0.5)^n\), \(P(X=k)=P(X=n-k)\). b) \(P(X\ge4)=0.34375\), and symmetry shows that this equals \(P(X\le2)\).

- First find the probability that all \(20\) LEDs in one strip work. - Use that strip-level probability in a second binomial model. - Keep the unrounded strip probability in part b).

1. A strip is completely functional with probability \(p=(0.98)^{20}\approx0.6676.\) 2. If \(Y\) counts functional strips among \(15\), then \(Y\sim\operatorname{Bin}(15,p)\). 3. \(P(Y=12)=\binom{15}{12}p^{12}(1-p)^3\approx0.1310.\)

a) \((0.98)^{20}\approx0.6676\). b) \(\binom{15}{12}[(0.98)^{20}]^{12}[1-(0.98)^{20}]^3\approx0.1310\).

- First calculate the one-packet success probability. - Then treat each packet as one Bernoulli trial. - Round only after the second-stage calculation is complete.

1. For one packet, \(p=P(X\ge11)=\binom{12}{11}(0.9)^{11}(0.1)+(0.9)^{12}\approx0.6590\). 2. Let \(Y\sim\operatorname{Bin}(5,p)\). 3. \(P(Y\le2)=\sum_{k=0}^{2}\binom{5}{k}p^k(1-p)^{5-k}\approx0.2214\).

a) \(0.6590\). b) \(0.2214\).

- Use the same binomial model for all three parts. - Part b is a lower-tail cumulative probability. - Part c is an inclusive integer interval.

1. \(X\sim\operatorname{Bin}(200,0.04)\). 2. \(P(X=8)\approx0.1425\). 3. \(P(X\le10)\approx0.8200\). 4. \(P(5\le X\le12)\approx0.8451\).

a) \(0.1425\). b) \(0.8200\). c) \(0.8451\).

- Translate each verbal condition to integer bounds. - Use a complement for the upper tail if convenient. - Check carefully whether a value on the boundary belongs to the event.

1. \(X\sim\operatorname{Bin}(30,0.25)\). 2. \(P(X>10)\approx0.1057\). 3. The second event is \(5\le X\le9\), with probability approximately \(0.7055\).

a) \(0.1057\). b) \(0.7055\).

- Use the stated binomial parameters. - Translate “at most” and “at least” into cumulative tails. - State all four binomial conditions in context.

1. Let \(X\sim\operatorname{Bin}(30,0.85)\). 2. \(P(X\le22)\approx0.0698\). 3. \(P(X\ge28)\approx0.1514\). 4. A binomial model requires fixed \(n=30\), two relevant outcomes, independent trials, and constant \(p=0.85\).

1. \(0.0698\). 2. \(0.1514\). 3. Fixed \(n\), two outcomes, independence, and constant \(p\).

- Use the requested binomial model for the numerical parts. - The expected count is \(np\). - In part 3, ask whether one package's error status can affect nearby packages.

1. Under the binomial model, \(X\sim\operatorname{Bin}(50,0.04)\), and \(P(X>3)\approx0.1391\). 2. \(E(X)=50(0.04)=2\), and \(P(X=2)\approx0.2762\). 3. Independence is questionable because one jam can create several nearby errors, making consecutive package outcomes dependent.

1. \(0.1391\). 2. Expected count \(2\); \(P(X=2)\approx0.2762\). 3. Independence is not reasonable if a single jam affects several consecutive packages.

- Use one binomial model for all three events. - A strict upper tail can be found by complement. - Part 3 includes both endpoints.

1. \(X\sim\operatorname{Bin}(250,0.04)\). 2. \(P(X\le5)\approx0.0633\). 3. \(P(X>12)\approx0.2045\). 4. \(P(8\le X\le15)\approx0.7400\).

1. \(0.0633\). 2. \(0.2045\). 3. \(0.7400\).

- Translate “fewer than” to the correct integer endpoint. - Use cumulative binomial probabilities for the tails. - Use a point probability for exactly \(12\).

1. Let \(X\sim\operatorname{Bin}(80,0.15)\). 2. \(P(X<10)=P(X\le9)\approx0.2211\). 3. \(P(X\ge15)\approx0.2126\). 4. \(P(X=12)\approx0.1240\).

1. \(0.2211\). 2. \(0.2126\). 3. \(0.1240\).

- Convert the percentage threshold to a number of bulbs. - Does “more than” include the count exactly on the boundary? - Use an exact-count probability for A and an upper tail for B.

1. Let \(X\sim\operatorname{Bin}(80,0.025)\). 2. \(P(A)=P(X=2)\approx0.2741\). 3. Five percent of \(80\) is \(4\), so event B is \(X>4\). 4. \(P(B)=P(X\ge5)\approx0.0504\).

A: \(0.2741\). B: \(0.0504\).

- Use the same binomial model for both events. - “At most \(5\)” is a lower tail. - Event D includes both endpoints.

1. \(X\sim\operatorname{Bin}(60,0.15)\). 2. \(P(C)=P(X\le5)\approx0.0968\). 3. \(P(D)=P(8\le X\le12)\approx0.5891\).

C: \(0.0968\). D: \(0.5891\).

- Use separate binomial models for the two sample sizes. - “At least \(14\)” includes \(14\) and \(15\). - A probability less than \(1\) cannot support a guarantee.

1. For part a), \(X\sim\operatorname{Bin}(15,0.90)\), so \(P(X\ge14)\approx0.5490\). 2. For part b), \(Y\sim\operatorname{Bin}(11,0.90)\), and \(P(Y\ge10)\approx0.6974\). 3. Since this probability is below \(1\), buying \(10\%\) extra does not guarantee \(10\) plants.

a) \(0.5490\). b) \(0.6974\); the guarantee claim is false.

- Use one binomial model for the whole problem. - Compare the two point probabilities directly in part b). - Do not infer likelihood only from closeness to the mean without calculation.

1. Let \(X\sim\operatorname{Bin}(50,0.05)\). 2. \(P(X\le3)\approx0.7604\). 3. \(P(X=5)\approx0.0658\), while \(P(X=0)\approx0.0769\). 4. Since \(0.0658<0.0769\), the claim is false.

a) \(0.7604\). b) \(P(X=5)\approx0.0658\), \(P(X=0)\approx0.0769\); the claim is false.

- What conditions define a binomial experiment? - Could wind or weather affect several shots at once? - Could a previous result affect the athlete's mental state? - What does independence require about the probability of the next trial?

1. Each shot must have two relevant outcomes, hit or miss. The hit probability must remain \(0.80\) for all \(5\) shots, and the shots must be independent. 2. In competition, wind or other environmental conditions may affect several shots, and psychological pressure or fatigue may change performance over time. These effects can create dependence or a changing hit probability. 3. If the next-shot probability depends on whether the previous shot was a miss, the trials are not independent and the success probability is not constant. Therefore, the binomial model is not valid.

a) Two relevant outcomes per shot, a constant success probability \(p=0.80\), and independent shots. b) Weather, fatigue, or psychological effects may influence multiple shots or change the hit probability. c) Independence and constant \(p\) are violated because the next-shot probability depends on the previous result.

- “At least \(4\)” is efficiently handled with a complement. - The draws are not exactly independent because sampling is without replacement. - Ask when removing one sampled component changes the composition of the remaining population only negligibly.

1. Let \(X\) be the number of selected components with a quality seal. 2. Because the sample is small relative to the very large production lot, the success probability changes negligibly from draw to draw, so use \(X\approx\operatorname{Bin}(12,0.15)\). 3. Use a complement: \(P(X\ge4)=1-\sum_{k=0}^{3}\binom{12}{k}(0.15)^k(0.85)^{12-k}.\) 4. This gives \(P(X\ge4)\approx0.0922.\) 5. The approximation is reasonable when the sample is only a small fraction of the population, so sampling without replacement creates negligible dependence and the success probability remains nearly constant.

\(P(X\ge4)\approx0.0922\). The binomial approximation is reasonable because the sample is very small relative to the population, making the draws approximately independent with nearly constant success probability.

- Convert the blue central angle into a fraction of \(360^\circ\). - “At most \(1\)” includes \(0\) and \(1\). - Keep the exact expressions until the final rounding.

1. The probability of blue is the fraction of the full circle occupied by the blue section: \(p=\frac{108}{360}=0.3.\) 2. The number of trials is \(n=20\). 3. Let \(X\) be the number of blue results. Then \(P(X=0)=(0.7)^{20}=0.0007979\ldots\approx0.0008.\) 4. \(P(X\le1)=P(X=0)+P(X=1) =(0.7)^{20}+20(0.3)(0.7)^{19} =0.0076373\ldots\approx0.0076.\)

a) \(n=20\), the number of spins, and \(p=0.3\), the probability of blue on each spin. b) \(P(X=0)\approx0.0008\). c) \(P(X\le1)\approx0.0076\).

- Decide whether each event specifies one sequence or only a total number of makes. - A binomial coefficient counts possible arrangements. - “At least” includes all qualifying exact counts. - A fixed sequence is found by multiplying probabilities in order.

1. Event \(A\) matches (4) because the binomial coefficient counts all arrangements of \(2\) makes and \(2\) misses. 2. Event \(B\) matches (1) because all \(4\) shots must be made. 3. Event \(C\) matches (2) because the required sequence is miss, miss, miss, make. 4. Event \(D\) matches (3) because one specific sequence of \(2\) misses followed by \(2\) makes is required. 5. Event \(E\) matches (5) because at least \(3\) makes means \(3\) or \(4\) makes. 6. \(P(E)=\binom{4}{3}(0.75)^3(0.25)+(0.75)^4=0.73828125.\)

a) \(A\to(4)\): all arrangements of two makes; \(B\to(1)\): four makes in a row; \(C\to(2)\): three misses, then one make; \(D\to(3)\): one fixed miss-miss-make-make order; \(E\to(5)\): either three or four makes. b) \(P(E)=0.73828125\).

- A first success in a specified position requires failures before that position and a success at that position. - “At most \(1\)” includes \(0\) and \(1\). - In part c), compare the factors in the two fixed-sequence probabilities.

1. For the first defect to occur in position \(3\), the first \(2\) components must be nondefective and the third must be defective. The remaining \(7\) outcomes are unrestricted. 2. Therefore, \(P=(0.96)^2(0.04)=0.036864.\) 3. For part b), let \(X\sim\operatorname{Bin}(10,0.04)\). 4. \(P(X\le1)=(0.96)^{10}+10(0.04)(0.96)^9\approx0.9419.\) 5. Each event in part c) has exactly one defective outcome and nine nondefective outcomes in a specified order. 6. Both probabilities are \((0.04)(0.96)^9\). Changing the position does not change the product.

a) \(0.036864\). b) \(P(X\le1)\approx0.9419\). c) Both events have probability \((0.04)(0.96)^9\), so they are equally likely.

- Model the number germinating with a binomial random variable. - Use the binomial-mode rule involving \((n+1)p\). - “At least \(24\)” includes \(24\) through \(30\).

1. Let \(X\sim\operatorname{Bin}(30,0.85)\). 2. Since \((n+1)p=31(0.85)=26.35\) is not an integer, the unique binomial mode is \(\lfloor26.35\rfloor=26.\) 3. \(P(X\ge24)=\sum_{k=24}^{30}\binom{30}{k}(0.85)^k(0.15)^{30-k}\approx0.8474.\)

a) \(26\) seeds. b) \(P(X\ge24)\approx0.8474\).

- Translate “fewer than \(50\)” into attainable integer values. - Use cumulative probabilities to calculate the interval. - The prompt already supplies fixed \(n\) and two relevant outcomes; identify the two remaining binomial conditions.

1. The binomial parameters are \(n=50\) and \(p=0.94\). 2. \(P(X=48)=\binom{50}{48}(0.94)^{48}(0.06)^2\approx0.2262.\) 3. \(P(X\le45)\approx0.1794.\) 4. \(P(46\le X<50)=P(46\le X\le49)\approx0.7753.\) 5. The additional modeling conditions are independent battery outcomes and the same success probability \(0.94\) for every battery.

a) (1) \(0.2262\); (2) \(0.1794\); (3) \(0.7753\). b) Independence and a constant success probability \(p=0.94\).

- Determine the largest attendance count that fits. - Counting no-shows can make the event easier to express. - Use a cumulative binomial probability or its complement.

1. Let \(X\) be the number of people who attend. Then \(X\sim\operatorname{Bin}(50,0.90)\). 2. Everyone has a seat when \(X\le45\). 3. Equivalently, let \(Y\) be the number of no-shows. Then \(Y\sim\operatorname{Bin}(50,0.10)\), and everyone has a seat when \(Y\ge5\). 4. \(P(Y\ge5)=1-P(Y\le4)\approx0.5688.\)

\(P(\text{everyone has a seat})\approx0.5688\).

- Translate “more than \(1\)” into a complement of \(0\) or \(1\). - Use the appropriate defect probability in each part. - Keep the sample size fixed at \(20\).

1. Let \(X\) be the number of cracked eggs in the sample. Rejection occurs when \(X>1\). 2. For part a), \(X\sim\operatorname{Bin}(20,0.02)\), so \(P(X>1)=1-\left[(0.98)^{20}+20(0.02)(0.98)^{19}\right]\approx0.0599.\) 3. For part b), \(X\sim\operatorname{Bin}(20,0.01)\), so \(P(X>1)=1-\left[(0.99)^{20}+20(0.01)(0.99)^{19}\right]\approx0.0169.\)

a) \(0.0599\). b) \(0.0169\).

- Interpret \(0.9\) as the success probability and use the exponents to identify counts. - Use the binomial coefficient or total exponents to identify \(n\). - An expression beginning with \(1-\) represents a complementary event. - Identify which exact probabilities are included in each sum.

1. In every expression, a success means that a seed germinates, with \(p=0.90\). 2. Part a) has \(n=20\) and \(k=18\). It represents planting \(20\) seeds and exactly \(18\) germinating. 3. In part b), the terms in brackets are \(P(X=0)+P(X=1)\) for \(n=10\). The complement is \(P(X\ge 2)\). It represents planting \(10\) seeds and at least \(2\) germinating. 4. In part c), the sum is \(P(X\le 16)\) for \(n=20\). The complement is \(P(X\ge 17)\). It represents planting \(20\) seeds and at least \(17\) germinating.

a) Plant \(20\) seeds; exactly \(18\) germinate. b) Plant \(10\) seeds; at least \(2\) germinate. c) Plant \(20\) seeds; at least \(17\) germinate.

- First find the probability that one figurine is flawless. - Use independence within a set to find the all-flawless set probability. - Treat each set as one Bernoulli trial in the second-stage model.

1. The probability that one figurine is flawless is \(0.60\). 2. A set contains only flawless figurines with probability \(p_S=(0.60)^5=0.07776.\) 3. Let \(X\) be the number of all-flawless sets among the \(20\) sets. Then \(X\sim\operatorname{Bin}(20,0.07776)\). 4. \(P(X\le2)=\sum_{k=0}^{2}\binom{20}{k}(0.07776)^k(0.92224)^{20-k}\approx0.7997.\)

\(P(X\le2)\approx0.7997\).

- First find the probability that one four-shot set is successful. - Treat each set as one Bernoulli trial after finding that set-level success probability. - After setting up the second count, decide which possible values meet the final condition.

1. The probability that one set is successful is \(p_S=\binom{4}{3}(0.70)^3(0.30)+(0.70)^4=0.6517\). 2. Let \(Y\) be the number of successful sets among the \(10\) sets. Then \(Y\sim\operatorname{Bin}(10,0.6517)\). 3. Therefore, \(P(Y\ge9)=\binom{10}{9}(0.6517)^9(0.3483)+(0.6517)^{10}\approx0.0877\).

\(P(Y\ge9)\approx0.0877\).

- Use \(np\) for the expected count. - Apply the binomial point formula for exactly \(18\). - “At least \(19\)” includes \(19\) and \(20\).

1. The germination probability is \(p=0.80\). 2. For \(450\) seeds, the expected number is \(450(0.80)=360\). 3. For \(20\) seeds, let \(X\sim\operatorname{Bin}(20,0.80)\). 4. \(P(X=18)=\binom{20}{18}(0.80)^{18}(0.20)^2\approx0.1369.\) 5. \(P(X\ge19)=\binom{20}{19}(0.80)^{19}(0.20)+(0.80)^{20}\approx0.0692.\)

a) \(360\) seedlings. b) \(0.1369\). c) \(0.0692\).

- Identify the binomial parameters from the survey. - Express “at least” using a complement. - Find an inclusive interval probability by subtracting cumulative probabilities with the correct endpoints.

1. Model the count with \(X\sim\operatorname{Bin}(150,0.35)\). 2. \(P(X\ge60)=1-P(X\le59)\approx0.1160.\) 3. \(P(45\le X\le60)=P(X\le60)-P(X\le44)\approx0.8295.\)

a) \(0.1160\). b) \(0.8295\).

- For part a), identify the required outcome of each of the first twelve tests. - Use the binomial mean and standard-deviation formulas in part b). - Convert the continuous one-standard-deviation interval to attainable integer counts.

1. For the twelfth chip to be the first defective chip, the first \(11\) chips must be nondefective and the twelfth must be defective: \(P=(0.92)^{11}(0.08)\approx0.0320.\) 2. Since \(X\sim\operatorname{Bin}(400,0.08),\) its mean is \(\mu=400(0.08)=32\) and its standard deviation is \(\sigma=\sqrt{400(0.08)(0.92)}=\sqrt{29.44}\approx5.43.\) 3. The interval \([\mu-\sigma,\mu+\sigma]\) is approximately \([26.57,37.43]\), so the included integer values are \(27\) through \(37\). 4. Therefore, \(P(27\le X\le37)\approx0.6898.\)

a) \((0.92)^{11}(0.08)\approx0.0320\). b) \(P(27\le X\le37)\approx0.6898\).

- Use the binomial formula for one exact value. - The mode is near the mean but is determined by the binomial-mode rule. - Evaluate \((n+1)p\).

1. \(X\sim\operatorname{Bin}(120,0.92)\). 2. \(P(X=110)=\binom{120}{110}(0.92)^{110}(0.08)^{10}\approx0.1295.\) 3. Since \((n+1)p=121(0.92)=111.32\) is not an integer, the unique mode is \(111\).

a) \(0.1295\). b) \(111\) seeds.

- Interpret \(np\) as a long-run average count. - Keep the exact square-root form of the standard deviation until the final approximation. - Convert the one-standard-deviation interval to attainable integer counts.

1. Since \(X\sim\operatorname{Bin}(500,0.02)\), \(\mu=np=10\). Over many groups of \(500\) healthy people, the average number of false-positive results would be about \(10\) per group. 2. \(\sigma=\sqrt{np(1-p)}=\sqrt{9.8}\approx3.13\). 3. The interval \([\mu-\sigma,\mu+\sigma]\) is approximately \([6.87,13.13]\), so the integer values are \(7\) through \(13\). 4. \(P(7\le X\le13)\approx0.7391\).

a) \(\mu=10\) and \(\sigma=\sqrt{9.8}\approx3.13\). The mean is the long-run average false-positive count per group of \(500\) healthy people. b) \(0.7391\).

- Use the binomial mean and standard-deviation formulas. - Translate the two-standard-deviation condition into an interval. - Keep only attainable integer values inside that interval.

1. Since \(Y\sim\operatorname{Bin}(300,0.10)\), \(\mu=30\). 2. \(\sigma=\sqrt{300(0.10)(0.90)}=\sqrt{27}=3\sqrt{3}\approx5.20\). 3. The interval \([\mu-2\sigma,\mu+2\sigma]\) is approximately \([19.61,40.39]\), so the integer values are \(20\) through \(40\). 4. \(P(20\le Y\le40)\approx0.9575\).

a) \(\mu=30\) and \(\sigma=3\sqrt{3}\approx5.20\). b) \(0.9575\).

- What happens to the shape of a binomial distribution when \(p=0.5\)? - Where is the center of symmetry? - Find the expected value and examine the adjacent integers.

1. For a binomial distribution, if \((n+1)p\) is an integer \(m\), the two modes are \(m-1\) and \(m\). 2. Here, \((25+1)\cdot0.5=13\), so the two modes are \(12\) and \(13\). 3. Because \(p=0.5\), the distribution is symmetric about \(np=12.5\). The values \(12\) and \(13\) are equally far from \(12.5\), so \(P(Y=12)=P(Y=13)\).

a) Since \((n+1)p=13\) is an integer, the binomial distribution has equal maximum probabilities at the adjacent values \(13-1\) and \(13\). b) The values are \(12\) and \(13\). Because \(p=0.5\), the distribution is symmetric about \(12.5\), so these equally spaced values have equal probabilities.

- Compute the recognition cutoff using the original process parameters. - Convert the continuous cutoff to the correct integer-count event. - For the final probability, use the explicitly stated independent new-process outcomes with delayed-call probability \(0.08\). - The requested event is the complement of the recognition event.

1. Under the original rate, \(n=120\) and \(p_0=0.15\). The mean is \(\mu_0=18\), and the standard deviation is \(\sigma_0=\sqrt{120(0.15)(0.85)}\approx3.9115\). 2. Success requires \(X<\mu_0-\sigma_0\approx14.0885\). Since \(X\) is an integer, success means \(X\le14\). 3. Under the new staffing tool, the outcomes are independent with delayed-call probability \(0.08\), so \(X\sim\operatorname{Bin}(120,0.08)\). 4. Failure to recognize the improvement means \(X\ge15\), so \(P(X\ge15)\approx0.0564\).

\(0.0564\).

- Build the recognition cutoff using the original process. - Translate the cutoff to the correct integer event. - For the final probability, use the explicitly stated independent new-process outcomes with defect probability \(0.04\). - The requested event is the complement of recognizing the improvement.

1. Under the original process, \(\mu_0=250(0.08)=20\), and \(\sigma_0=\sqrt{250(0.08)(0.92)}\approx4.2895\). 2. Recognition requires \(X\le20-1.5(4.2895)\approx13.5657\). Since \(X\) is integer-valued, recognition occurs for \(X\le13\). 3. Under the new production process, the outcomes are independent with defect probability \(0.04\), so \(X\sim\operatorname{Bin}(250,0.04)\). 4. Therefore, \(P(\text{not recognized})=P(X\ge14)\approx0.1310\).

\(0.1310\).

- How many possible values can \(X\) take when \(n=21\)? - For a symmetric distribution with an even number of possible values, compare the sums over the lower and upper halves. - Pay attention to the difference between \(<\) and \(\le\). - Use the result from part a) to simplify part b).

1. Because \(p=0.5\), the distribution is symmetric about \(10.5\), so \(P(X=k)=P(X=21-k)\). 2. The values \(0\) through \(10\) pair with the values \(21\) through \(11\). Therefore, \(P(X\le 10)=P(X\ge 11)\). These two complementary events have equal probability, so each has probability \(0.5\). 3. \(P(X=10)=\binom{21}{10}(0.5)^{21}\approx 0.1682\). 4. Therefore, \(P(X<10)=P(X\le 10)-P(X=10)\approx 0.5-0.1682=0.3318\).

a) Symmetry pairs each outcome from \(0\) through \(10\) with an equally likely outcome from \(11\) through \(21\). The two halves are complementary, so \(P(X\le 10)=0.5\). b) \(P(X=10)\approx 0.1682\) \(P(X<10)\approx 0.3318\)

- Divide the number of observed defects by the total sample size. - Distinguish a probability estimated from data from one derived from a specified model. - Use the binomial point-probability formula in part 2.

1. The relative frequency is \(\frac{15}{500}=0.03,\) or \(3\%\). Using observed relative frequency as an estimate is an experimental or empirical probability approach. 2. Let \(X\sim\operatorname{Bin}(20,0.02)\). Then \(P(X=1)=\binom{20}{1}(0.02)(0.98)^{19}\approx0.2725.\) 3. This second calculation uses a theoretical, model-based probability because it begins with a specified probability model and parameter.

1. \(0.03\), or \(3\%\); experimental or empirical probability. 2. \(\binom{20}{1}(0.02)(0.98)^{19}\approx0.2725\); theoretical, model-based probability.

- Determine a different one-spin success probability for each event. - Use complements when they shorten the cumulative calculation. - Independence means past spins do not change the probability of the next spin.

1. For part a)(1), the multiples of \(3\) are \(3,6,9,12\), so \(p=\frac{1}{3}\). Thus, \(P(X\ge5)\approx0.9538\). 2. For part a)(2), \(p=\frac{1}{12}\), so \(P(Y>2)\approx0.3467\). 3. For part a)(3), the prime labels are \(2,3,5,7,11\), so \(p=\frac{5}{12}\), and \(P(Z\le10)\approx0.5180\). 4. The spins are independent, so previous results do not change the next-spin probability. Therefore, \(P(\text{12 on spin 26})=\frac{1}{12}\).

a) (1) \(0.9538\); (2) \(0.3467\); (3) \(0.5180\). b) \(\frac{1}{12}\).

- Use the binomial formulas for center and spread. - Translate the one-standard-deviation interval into attainable integer counts. - Use cumulative probabilities for the inclusive interval.

1. Since \(X\sim\operatorname{Bin}(100,0.60)\), \(\mu=60\) and \(\sigma=\sqrt{100(0.60)(0.40)}=\sqrt{24}=2\sqrt{6}\approx4.90\). 2. The interval \([\mu-\sigma,\mu+\sigma]\) is approximately \([55.10,64.90]\), so the included integer values are \(56\) through \(64\). 3. \(P(56\le X\le64)\approx0.6416\).

1. \(\mu=60\) and \(\sigma=2\sqrt{6}\approx4.90\). 2. \(0.6416\).

- Start from the binomial model and find its mean and standard deviation. - The interval endpoints need not be integers, but the random variable can take only integer counts. - After identifying the included counts, evaluate the corresponding binomial interval probability.

1. Since \(X\sim\operatorname{Bin}(80,0.2)\), its mean and standard deviation are \(\mu=80(0.2)=16\) and \(\sigma=\sqrt{80(0.2)(0.8)}=\sqrt{12.8}\approx3.578\). 2. Thus, \(2\sigma\approx7.155\), so the continuous interval is approximately \([8.845,23.155]\). 3. Because \(X\) is an integer-valued count, the values within the interval are \(9,10,\ldots,23\). 4. Therefore, \(P(|X-\mu|\le2\sigma)=P(9\le X\le23)=P(X\le23)-P(X\le8)\approx0.9652\).

a) Approximately \([8.845,23.155]\), containing the integer counts \(9\) through \(23\). b) \(P(9\le X\le23)\approx0.9652\).

- Decide whether each event specifies an order or only a total number of successes. - Break “at least” into the qualifying exact counts. - Use no binomial coefficient for a fully specified sequence.

1. Let \(X\sim\operatorname{Bin}(12,0.85)\). 2. \(P(X=9)=\binom{12}{9}(0.85)^9(0.15)^3\approx0.1720.\) 3. \(P(X\ge11)=P(X=11)+P(X=12)\approx0.4435.\) 4. The third event specifies one particular ordered sequence, so \(P=(0.15)^3(0.85)^9\approx0.000782.\)

(1) \(0.1720\). (2) \(0.4435\). (3) \((0.15)^3(0.85)^9\approx0.000782\).

- Find the expected number of positive responses. - Translate “differs by at most \(4\)” into an inclusive integer interval. - Evaluate the interval with cumulative binomial probabilities.

1. Since \(X\sim\operatorname{Bin}(50,0.40)\), \(\mu=np=20\). 2. A difference of at most \(4\) means \(16\le X\le24\). 3. Therefore, \(P(16\le X\le24)\approx0.8067\).

\(0.8067\).

- Find the probability of one fully specified sequence. - Ask whether an apparently mixed pattern changes the probability of that exact sequence. - Count the sequences containing exactly four tails. - Distinguish one outcome sequence from an event containing many sequences.

1. Each exact sequence of eight fair coin flips has probability \((0.5)^8=\frac{1}{256}\approx0.0039\). Therefore, Sequences 1 and 2 are equally likely. 2. The flips are independent, and heads and tails have equal probability. The visual pattern of an exact sequence does not change its probability. 3. For exactly four tails, \(P(X=4)=\binom{8}{4}(0.5)^4(0.5)^4=\frac{70}{256}\approx0.2734\). 4. The event “exactly four tails” contains \(\binom{8}{4}=70\) equally likely sequences, while “eight tails” contains only one sequence.

a) Each sequence has probability \(\frac{1}{256}\approx0.0039\). b) Every exact eight-flip sequence is equally likely because the flips are independent and the coin is fair. c) \(P(X=4)=\frac{70}{256}\approx0.2734\). The event contains \(70\) sequences, compared with one sequence for eight tails.

- Write the probabilities of a make and a miss. - For a specified sequence, multiply the probabilities in order. - Ask whether changing the order of the same factors changes their product. - Interpret \(\binom{n}{k}\) as the number of sequences with \(k\) successes.

1. The make probability is \(p=0.8\), and the miss probability is \(0.2\). 2. \(P(S_1)=(0.8)^4=0.4096\), and \(P(S_2)=(0.8)^3(0.2)=0.1024\). 3. The statement is true. Every sequence with exactly three makes and one miss has probability \((0.8)^3(0.2)\), regardless of the order of the factors. For example, \(P(F, M, M, M)=0.2\cdot(0.8)^3=0.1024=P(M, F, M, M)\). 4. There are \(\binom{4}{3}=4\) sequences with exactly three makes, so \(P(X=3)=\binom{4}{3}(0.8)^3(0.2)=0.4096\).

a) \(P(S_1)=0.4096\); \(P(S_2)=0.1024\). b) The statement is true. Each sequence with three makes and one miss has probability \((0.8)^3(0.2)=0.1024\). c) \(P(X=3)=0.4096\).

- When can sampling without replacement be approximated by independent trials? - Look for language indicating that success chances change over time. - Can every trial be classified into success or failure?

1. In a), a binomial model is a reasonable approximation because the city population is very large compared with the sample. The ownership probability can be treated as approximately constant at \(p=0.04\), with \(n=100\). The random variable \(X\) counts electric-vehicle owners in the sample. 2. In b), fatigue causes the success probability to change over time. A binomial model is not appropriate. 3. In c), the spins are independent, and the probability of yellow is constant at \(p=\frac{1}{3}\). This is binomial with \(n=20\), and \(X\) counts yellow outcomes.

a) Yes, as an approximation: \(n=100\), \(p=0.04\), and \(X\) is the number of electric-vehicle owners in the sample. b) No; fatigue makes the success probability nonconstant. c) Yes: \(n=20\), \(p=\frac{1}{3}\), and \(X\) is the number of yellow outcomes.

- Use the binomial formulas for the mean and standard deviation. - Find the continuous one-standard-deviation interval before identifying the integer counts it contains. - Sum the binomial probabilities over those attainable counts.

1. Since \(X\sim\operatorname{Bin}(400,0.05)\), \(\mu=np=20.\) 2. The standard deviation is \(\sigma=\sqrt{np(1-p)}=\sqrt{19}\approx4.36.\) 3. The continuous interval is \([20-\sqrt{19},20+\sqrt{19}]\approx[15.64,24.36].\) Therefore, the attainable integer values in the interval are \(16\) through \(24\). 4. Hence, \(P(\mu-\sigma\le X\le\mu+\sigma) =P(16\le X\le24) \approx0.6990.\)

a) \(\mu=20\), \(\sigma=\sqrt{19}\approx4.36\). b) \(P(16\le X\le24)\approx0.6990\).

- Use the binomial mean and standard-deviation formulas. - Translate the one-standard-deviation interval into attainable integer counts. - The requested event is outside that interval.

1. Since \(X\sim\operatorname{Bin}(150,0.64)\), \(\mu=150(0.64)=96.\) 2. \(\sigma=\sqrt{150(0.64)(0.36)}=\sqrt{34.56}=\frac{12\sqrt6}{5}\approx5.88.\) 3. Values within one standard deviation satisfy approximately \(90.12\le X\le101.88\), so the included integers are \(91\) through \(101\). 4. Therefore, the event more than one standard deviation from the mean is \(X\le90\) or \(X\ge102\), with probability \(1-P(91\le X\le101)\approx0.3495.\)

a) \(\mu=96\) and \(\sigma=\frac{12\sqrt6}{5}\approx5.88\). b) \(0.3495\).

- What fraction of the four choices is correct for each question? - “At most \(1\)” is a cumulative probability. - Use a complement to obtain \(P(X>2)\), then compare it directly with \(P(X=2)\).

1. Let \(X\sim\operatorname{Bin}(8,0.25)\). 2. \(P(X=2)\approx0.3115\). 3. \(P(X\le1)\approx0.3671\). 4. \(P(X>2)=1-P(X\le2)\approx0.3215\). 5. Since \(0.3215>0.3115\), more than two correct answers is slightly more likely.

a) \(0.3115\). b) \(0.3671\). c) More than \(2\) correct answers is more likely: \(0.3215\) versus \(0.3115\).

- Model the defective count with one binomial random variable. - A complement is efficient for “at least \(2\).” - In part c), calculate the two event probabilities before comparing them.

1. Let \(X\sim\operatorname{Bin}(12,0.10)\). 2. For part a), \(P(X=0)=(0.90)^{12}\approx0.2824\). 3. For part b), \(P(X\ge2)=1-[P(X=0)+P(X=1)]\approx0.3410\). 4. For part c), \(P(X=1)=12(0.10)(0.90)^{11}\approx0.3766\). 5. Also, \(P(X>2)=1-P(X\le2)\approx0.1109\). Since \(0.3766>0.1109\), exactly one defective figurine is more likely.

a) \(0.2824\). b) \(0.3410\). c) Exactly \(1\) defective figurine is more likely: \(0.3766\) versus \(0.1109\).

- Determine the probability of a correct answer with two choices. - Model the questions as independent Bernoulli trials. - For \(6\) questions, which correct-answer counts mean more correct than incorrect? - Use the binomial formula for each exact count.

1. Let \(X\sim\operatorname{Bin}(6,0.5)\). 2. For part (1), \(P(X=3)=\binom{6}{3}(0.5)^6=0.3125\). 3. For part (2), \(P(X=5)=\binom{6}{5}(0.5)^6=0.09375\). 4. More answers are correct than incorrect when \(X=4,5,\) or \(6\). Thus, \(P(X\ge 4)=0.234375+0.09375+0.015625=0.34375\).

(1) The probability is \(0.3125\), or \(31.25\%\). (2) The probability is \(0.09375\), or \(9.375\%\). (3) The probability is \(0.34375\), or \(34.375\%\).

- Use the same binomial structure with two different success probabilities. - List the possible counts that meet the upper-tail condition before calculating. - Compare the probabilities after changing only \(p\).

1. For \(p=0.70\), \(P(X=2)=\binom{4}{2}(0.70)^2(0.30)^2=0.2646,\) and \(P(X\ge3)=\binom{4}{3}(0.70)^3(0.30)+(0.70)^4=0.6517.\) 2. For \(p=0.75\), \(P(X=2)=\binom{4}{2}(0.75)^2(0.25)^2=0.2109375\approx0.2109,\) and \(P(X\ge3)=\binom{4}{3}(0.75)^3(0.25)+(0.75)^4=0.73828125\approx0.7383.\) 3. Raising the success probability shifts probability toward larger success counts: exactly two makes becomes less likely, while at least three makes becomes more likely.

a) \(P(X=2)=0.2646\); \(P(X\ge3)=0.6517\). b) \(P(X=2)\approx0.2109\); \(P(X\ge3)\approx0.7383\). Exactly two makes becomes less likely, while at least three makes becomes more likely.

- Translate each verbal condition into a condition on the number germinating. - Since the count is an integer, think carefully about the boundary in “more than.” - Translate the last part into a statement about germinating seeds, using the fixed packet size.

1. Let \(X\sim\operatorname{Bin}(12,0.80)\), where \(X\) is the number of seeds that germinate. 2. \(P(X=10)\approx0.2835\). 3. \(P(X>10)=P(X=11)+P(X=12)\approx0.2749\). 4. At most \(2\) seeds fail to germinate means at least \(10\) germinate, so \(P(X\ge10)\approx0.5583\).

(1) \(0.2835\). (2) \(0.2749\). (3) \(0.5583\).

- Identify the two exact success counts included in the event. - The events cannot occur together, so their probabilities can be added. - Use the binomial point-probability formula for each count.

1. Let \(X\sim\operatorname{Bin}(10,0.25)\). 2. The two events are mutually exclusive, so \(P(X=3\text{ or }X=4)=P(X=3)+P(X=4).\) 3. \(P(X=3)=\binom{10}{3}(0.25)^3(0.75)^7,\) and \(P(X=4)=\binom{10}{4}(0.25)^4(0.75)^6.\) 4. Therefore, \(P(X=3\text{ or }X=4)\approx0.3963.\)

\(0.3963\).

- First derive the success probability from the two-dice sample space. - Translate “fewer than \(2\)” into the qualifying binomial counts. - Keep the exact fractions until the final rounding.

1. A sum of \(11\) occurs with \((5, 6)\) or \((6, 5)\), so the one-trial success probability is \(p=\frac{2}{36}=\frac{1}{18}\). 2. Let \(X\sim\operatorname{Bin}(8,\frac{1}{18})\). 3. Fewer than \(2\) occurrences means \(X=0\) or \(X=1\). 4. \(P(X<2)=\left(\frac{17}{18}\right)^8+8\left(\frac{1}{18}\right)\left(\frac{17}{18}\right)^7\approx0.9309\).

\(0.9309\).

- Use the binomial mode rule involving \((n+1)p\). - The two ranges in part b) do not overlap. - Evaluate the lower tail and upper tail separately, then add them.

1. \(X\sim\operatorname{Bin}(20,0.60)\). 2. Since \((n+1)p=21(0.60)=12.6\) is not an integer, the unique mode is \(\lfloor12.6\rfloor=12\). 3. The events \(X\le8\) and \(X\ge16\) are disjoint, so their probabilities add. 4. Therefore, \(P(X\le8\text{ or }X\ge16)=P(X\le8)+P(X\ge16)\approx0.1075\).

a) \(12\). b) \(0.1075\).

- Use the binomial mode rule involving \((n+1)p\). - The two tail events are disjoint. - Use a complement for the upper tail and add the two tail probabilities.

1. \(X\sim\operatorname{Bin}(20,0.20)\). 2. Since \((n+1)p=21(0.20)=4.2\) is not an integer, the unique mode is \(\lfloor4.2\rfloor=4\). 3. The events in part b) are disjoint. \(P(X\le1)=(0.8)^{20}+\binom{20}{1}(0.2)(0.8)^{19}\approx0.0692\). 4. Also, \(P(X\ge8)=1-P(X\le7)\approx0.0321\). 5. Therefore, \(P(X\le1\text{ or }X\ge8)\approx0.1013\).

a) \(4\). b) \(0.1013\).

- Distinguish a theoretical probability from the outcome of a small set of trials. - Decide whether one spin affects the next spin. - Think about what happens to relative frequency as the number of trials grows. - Compare the binomial probabilities for exactly \(10\) and exactly \(11\) successes.

1. Statement (1) is false. A probability of \(0.25\) does not guarantee an exact result in a small number of trials. The probability of exactly one red result in \(4\) spins is \(\binom{4}{1}(0.25)(0.75)^3\approx 0.4219\). 2. Statement (2) is false. The spins are independent, so the probability of red remains \(0.25\) on every spin. 3. Statement (3) is true. By the law of large numbers, the relative frequency tends to become more stable near the theoretical probability as the number of trials increases. 4. Statement (4) is true. For \(X\sim\operatorname{Bin}(40,0.25)\), \(\frac{P(X=11)}{P(X=10)}=\frac{40-10}{11}\cdot\frac{0.25}{0.75}=\frac{10}{11}<1\). Therefore, \(P(X=10)>P(X=11)\).

(1) False (2) False (3) True (4) True

- Recall the mean of a binomial random variable. - Use independence to find the probability that all \(20\) chips are nondefective. - A sample proportion can differ from a population probability because of random variation. - A fair wager has an expected net gain of \(0\).

1. Statement (1) is true. For \(X\sim\operatorname{Bin}(100,0.05)\), \(E(X)=np=100\cdot0.05=5\). 2. Statement (2) is true. Each chip is nondefective with probability \(0.95\), so independence gives \(P(X=0)=(0.95)^{20}\). 3. Statement (3) is false. Random samples vary. In fact, \(P(X=3)=\binom{20}{3}(0.05)^3(0.95)^{17}\approx 0.0596\), so this result is possible under the stated model and does not by itself disprove the stated defect rate. 4. Statement (4) is true. If the bettor wins \(\$19\) when the chip is defective and loses \(\$1\) otherwise, the expected net gain is \(0.05\cdot19-0.95\cdot1=0\), so the wager is fair.

(1) True (2) True (3) False (4) True

- Count the multiples of \(3\) from \(1\) through \(10\). - Model the two independent spins with a binomial distribution. - Translate “the first or second result, or both” into a condition on \(X\). - Use the table or a complement to find the probability of at least one success.

1. The multiples of \(3\) are \(3\), \(6\), and \(9\), so the success probability is \(p=\frac{3}{10}=0.3\). 2. The two spins are independent, so \(X\) is binomial with \(n=2\) and \(p=0.3\). Thus, \(P(X=0)=(0.7)^2=0.49\), \(P(X=1)=\binom{2}{1}(0.3)(0.7)=0.42\), and \(P(X=2)=(0.3)^2=0.09\). 3. The event in part b) is \(X\ge1\). Therefore, \(P(X\ge1)=1-P(X=0)=1-0.49=0.51\). 4. Exactly one divisible result has probability \(P(X=1)=0.42\). Since \(0.51>0.42\), at least one divisible result is more likely.

a) <table> <tr><td>\(k\)</td><td>\(0\)</td><td>\(1\)</td><td>\(2\)</td></tr> <tr><td>\(P(X=k)\)</td><td>\(0.49\)</td><td>\(0.42\)</td><td>\(0.09\)</td></tr> </table> b) \(P(X\ge1)=0.51\) c) At least one result divisible by \(3\) is more likely because \(0.51>0.42\).

- Decide whether the guesses are independent. - Identify the number of trials and the success probability. - Use the binomial formula for an exact count. - For “at least,” identify the first and last whole-number counts that qualify.

1. Each guess is correct with probability \(0.5\), and the trials are independent, so \(X\sim\operatorname{Bin}(12,0.5)\). 2. \(P(X=9)=\binom{12}{9}\cdot (0.5)^{12}=\frac{220}{4096}\approx 0.0537\). 3. For at least \(10\) correct guesses, add the probabilities for \(10\), \(11\), and \(12\): \(P(X\ge 10)=\frac{66+12+1}{4096}=\frac{79}{4096}\approx 0.0193\).

a) \(P(X=9)=\frac{220}{4096}\approx 0.0537\), or about \(5.37\%\) b) \(P(X\ge 10)=\frac{79}{4096}\approx 0.0193\), or about \(1.93\%\)

- Does predicting first and second give an ordered or unordered outcome? - How many choices are left for second place after first place is chosen? - After finding the chance for one person, what model describes the count of correct predictions among the sixty people?

1. There are \(10\cdot9=90\) possible ordered pairs for first and second place. 2. Exactly one ordered pair is correct, so \(p=\frac{1}{90}\). 3. Let \(X\sim\operatorname{Bin}(60,\frac{1}{90})\) count the correct predictions. 4. \(P(X=1)=\binom{60}{1}\left(\frac{1}{90}\right)\left(\frac{89}{90}\right)^{59}\approx0.3448\).

a) \(p=\frac{1}{90}\). b) \(0.3448\).

- Use the binomial formula for one exact value. - “At most one” includes zero and one. - Interpret the binomial coefficient as a count of possible success-position choices.

1. \(X\sim\operatorname{Bin}(20,0.05)\). 2. \(P(X=2)=\binom{20}{2}(0.05)^2(0.95)^{18}\approx0.1887.\) 3. \(P(X\le1)=(0.95)^{20}+20(0.05)(0.95)^{19}\approx0.7358.\) 4. \(\binom{20}{2}=190.\) It counts the possible pairs of sample positions occupied by the two defective components.

a) \(0.1887\). b) \(0.7358\). c) \(190\); it counts the possible pairs of defect positions among the twenty sampled components.

- Use the binomial point formula in part a). - In part b), focus on choosing success positions rather than calculating a probability. - Use the complement of zero correct answers in part c).

1. \(X\sim\operatorname{Bin}(10,0.25)\). 2. \(P(X=3)=\binom{10}{3}(0.25)^3(0.75)^7\approx0.2503\). 3. To obtain exactly two correct answers, choose the two correct-answer positions among the ten questions. The number of patterns is \(\binom{10}{2}=45\). 4. \(P(X\ge1)=1-P(X=0)=1-(0.75)^{10}\approx0.9437\).

a) \(0.2503\). b) \(\binom{10}{2}=45\) patterns; the coefficient chooses the two positions containing correct answers. c) \(0.9437\).

- Write both point probabilities using the binomial formula. - Compare \(\binom{10}{2}\) with \(\binom{10}{8}\). - Use the fact that success and failure have the same probability when \(p=0.5\).

1. \(P(X=2)=\binom{10}{2}(0.5)^{10}=\frac{45}{1024}\approx0.0439\). 2. \(P(X=8)=\binom{10}{8}(0.5)^{10}=\frac{45}{1024}\approx0.0439\). 3. The identity \(\binom{10}{2}=\binom{10}{8}\), together with equal success and failure probabilities, makes the two binomial probability terms identical.

a) \(P(X=2)=P(X=8)=\frac{45}{1024}\approx0.0439\). b) The probabilities are equal because \(\binom{10}{2}=\binom{10}{8}\) and both probability products equal \((0.5)^{10}\).

- First find the probability of winning one round. - For “at least once,” consider the complement of no wins. - Model the rounds as independent Bernoulli trials.

1. There are \(\binom{18}{4}=3060\) possible sets of \(4\) numbers, so the probability of winning one round is \(p=\frac{1}{3060}\). 2. The probability of no wins in \(1000\) independent rounds is \(\left(1-\frac{1}{3060}\right)^{1000}\). 3. Therefore, the probability of at least one win is \(1-\left(\frac{3059}{3060}\right)^{1000}\approx 0.278809\).

The probability is approximately \(27.88\%\).

- Treat the named color as success and all other colors as failure. - The success probability is different for blue, yellow, and red. - Use a complement for more than two red results.

1. For blue, the one-spin success probability is \(10/20=0.5\), so \(P(\text{exactly }5\text{ blue})=\binom{15}{5}(0.5)^{15}\approx0.0916.\) 2. The probability of not landing on yellow is \(0.8\), so \(P(\text{no yellow})=(0.8)^{15}\approx0.0352.\) 3. For red, the one-spin success probability is \(6/20=0.3\). If \(X\sim\operatorname{Bin}(15,0.3)\), then \(P(X>2)=1-P(X\le2)\approx0.8732.\)

a) \(0.0916\). b) \(0.0352\). c) \(0.8732\).

- Identify the possible values of \(X\). - Decide what must stay the same across the shots to use a binomial model. - Apply the formula for exactly \(k\) successes. - Translate “at least \(3\)” into the relevant values of \(X\).

1. Independence and a constant success probability give \(X\sim\operatorname{Bin}(4,0.75)\). 2. Using \(P(X=k)=\binom{4}{k}(0.75)^k(0.25)^{4-k}\), the probabilities for \(k=0,1,2,3,4\) are \(\frac{1}{256}\), \(\frac{12}{256}\), \(\frac{54}{256}\), \(\frac{108}{256}\), and \(\frac{81}{256}\), respectively. 3. \(P(X\ge3)=P(X=3)+P(X=4)=\frac{108+81}{256}=\frac{189}{256}\approx0.7383\).

a) <table> <tr><th>\(k\)</th><th>\(0\)</th><th>\(1\)</th><th>\(2\)</th><th>\(3\)</th><th>\(4\)</th></tr> <tr><th>\(P(X=k)\)</th><td>\(0.0039\)</td><td>\(0.0469\)</td><td>\(0.2109\)</td><td>\(0.4219\)</td><td>\(0.3164\)</td></tr> </table> b) \(P(X\ge3)=\frac{189}{256}\approx0.7383\).

- What feature of the sampling makes the binomial model appropriate? - List every possible defect count from \(0\) through \(5\). - How would you summarize the average count over many such samples?

1. \(Y\sim\operatorname{Bin}(5,0.10)\). 2. For \(k=0,1,\ldots,5\), \(P(Y=k)=\binom{5}{k}(0.10)^k(0.90)^{5-k}\). 3. This gives probabilities \(0.59049\), \(0.32805\), \(0.0729\), \(0.0081\), \(0.00045\), and \(0.00001\), respectively. 4. \(E(Y)=np=5(0.10)=0.5\).

a) <table> <tr><th>\(k\)</th><th>\(0\)</th><th>\(1\)</th><th>\(2\)</th><th>\(3\)</th><th>\(4\)</th><th>\(5\)</th></tr> <tr><th>\(P(Y=k)\)</th><td>\(0.59049\)</td><td>\(0.32805\)</td><td>\(0.0729\)</td><td>\(0.0081\)</td><td>\(0.00045\)</td><td>\(0.00001\)</td></tr> </table> b) \(E(Y)=0.5\).

- Relate the number of losses to the number of wins in four spins. - Changing the focus from wins to losses swaps the success and failure probabilities. - Use the identity between complementary counts.

1. \(X\sim\operatorname{Bin}(4,\frac{1}{4})\), so for \(k=0,1,2,3,4\), \(P(X=k)=\frac{81}{256},\frac{108}{256},\frac{54}{256},\frac{12}{256},\frac{1}{256}\). 2. \(Y\sim\operatorname{Bin}(4,\frac{3}{4})\), giving the same probabilities in reverse order: \(\frac{1}{256},\frac{12}{256},\frac{54}{256},\frac{108}{256},\frac{81}{256}\). 3. Every outcome with \(k\) wins has exactly \(4-k\) losses, so \(P(X=k)=P(Y=4-k)\).

a) For \(k=0,1,2,3,4\): \(\frac{81}{256},\frac{108}{256},\frac{54}{256},\frac{12}{256},\frac{1}{256}\). b) For \(k=0,1,2,3,4\): \(\frac{1}{256},\frac{12}{256},\frac{54}{256},\frac{108}{256},\frac{81}{256}\). c) \(P(X=k)=P(Y=4-k)\); the distributions are reversals of each other.

- Express each event in terms of the number of failures. - Germinations plus failures always total ten. - Compare the two event descriptions before doing any arithmetic.

1. \(P(X\ge9)=\binom{10}{9}(0.9)^9(0.1)+(0.9)^{10}.\) 2. \(P(Y\le1)=(0.9)^{10}+10(0.1)(0.9)^9.\) 3. The two expressions are identical because the events are identical: at least \(9\) germinations means at most \(1\) failure.

a) \(\binom{10}{9}(0.9)^9(0.1)+(0.9)^{10}\). b) \((0.9)^{10}+10(0.1)(0.9)^9\). c) The events are the same event described in complementary counts.

- How many relevant outcomes can a Bernoulli trial have? - Can several outcomes be grouped into one success category? - A bulb must sprout before its flower color can be observed.

1. A Bernoulli trial has exactly two relevant categories, success and failure. Recording four separate outcomes does not meet that condition. 2. One definition is “success means the bulb sprouts.” Then \(p=1-0.10=0.90\). 3. Another definition is “success means the bulb blooms red.” The bulb must first sprout and then bloom red, so \(p=0.90\cdot0.60=0.54\).

a) The observation has four separate outcome categories rather than two. b) Example 1: success is “the bulb sprouts,” so \(p=0.90\). Example 2: success is “the bulb blooms red,” so \(p=0.54\).

- Use complements for events near the endpoints of the support. - “At most \(9\)” is the complement of exactly \(10\). - Preserve exact expressions when the probability is extremely close to \(0\) or \(1\).

1. \(X\sim\operatorname{Bin}(10,\frac{1}{8})\). 2. \(P(X\ge1)=1-\left(\frac{7}{8}\right)^{10}\approx0.7369\). 3. \(P(X>2)=1-P(X\le2)\approx0.1195\). 4. \(P(X\le9)=1-P(X=10)=1-\left(\frac{1}{8}\right)^{10}\). 5. \(P(X=10)=\left(\frac{1}{8}\right)^{10}\).

(1) \(0.7369\). (2) \(0.1195\). (3) \(1-\left(\frac{1}{8}\right)^{10}\). (4) \(\left(\frac{1}{8}\right)^{10}\).

- What two outcomes can each checked pad have, and why is the chance the same each time? - Decide which counts belong in each of the two events after making the table. - Think about whether a complement might shorten one calculation. - Keep unrounded probabilities through cumulative calculations.

1. \(X\sim\operatorname{Bin}(6,0.20)\), so \(P(X=k)=\binom{6}{k}(0.20)^k(0.80)^{6-k}\). 2. For \(k=0,1,\ldots,6\), the probabilities rounded to four decimals are \(0.2621,0.3932,0.2458,0.0819,0.0154,0.0015,0.0001\). 3. Using unrounded point probabilities, \(P(X\le2)=\sum_{k=0}^{2}\binom{6}{k}(0.20)^k(0.80)^{6-k}=0.90112\approx0.9011\). 4. Also, \(P(X\ge1)=1-P(X=0)=1-(0.80)^6=0.737856\approx0.7379\).

a) <table> <tr><th>\(k\)</th><th>\(0\)</th><th>\(1\)</th><th>\(2\)</th><th>\(3\)</th><th>\(4\)</th><th>\(5\)</th><th>\(6\)</th></tr> <tr><th>\(P(X=k)\)</th><td>\(0.2621\)</td><td>\(0.3932\)</td><td>\(0.2458\)</td><td>\(0.0819\)</td><td>\(0.0154\)</td><td>\(0.0015\)</td><td>\(0.0001\)</td></tr> </table> b) 1. \(0.9011\). 2. \(0.7379\).

- With \(p=0.5\), all length-ten answer patterns are equally likely. - Use symmetry about \(5\) in part c). - Preserve exact fractions instead of introducing unnecessary rounding.

1. \(X\sim\operatorname{Bin}(10,0.5)\). 2. \(P(X=5)=\binom{10}{5}(0.5)^{10}=\frac{252}{1024}=\frac{63}{256}\). 3. \(P(X\ge8)=\frac{\binom{10}{8}+\binom{10}{9}+\binom{10}{10}}{2^{10}}=\frac{56}{1024}=\frac{7}{128}\). 4. Because the distribution is symmetric about \(5\), \(P(X<5)=P(X>5)\). Hence \(P(X>5)=\frac{1-P(X=5)}{2}=\frac{193}{512}\). 5. Since \(\frac{193}{512}>\frac{63}{256}\), more than \(5\) correct answers is more likely.

a) \(\frac{63}{256}\). b) \(\frac{7}{128}\). c) More than \(5\) correct answers is more likely because \(P(X>5)=\frac{193}{512}>\frac{63}{256}=P(X=5)\).

- List all possible values for the number of ones in a \(3\)-bit sequence. - How many ones can an \(8\)-bit sequence have if it contains more zeros than ones? - When \(p=0.5\), each binomial term contains the same factor \((0.5)^n\).

1. For the shorter sequence, \(X_3\sim\operatorname{Bin}(3,0.5)\). The probabilities are \(P(X_3=0)=0.125\), \(P(X_3=1)=0.375\), \(P(X_3=2)=0.375\), and \(P(X_3=3)=0.125\). 2. For \(8\) bits, more zeros than ones means at most \(3\) ones. 3. Therefore, \(P(X_8\le3)=\frac{\binom{8}{0}+\binom{8}{1}+\binom{8}{2}+\binom{8}{3}}{2^8}=\frac{93}{256}\approx0.363281\).

a) <table> <tr><th>\(k\)</th><th>\(0\)</th><th>\(1\)</th><th>\(2\)</th><th>\(3\)</th></tr> <tr><th>\(P(X_3=k)\)</th><td>\(0.125\)</td><td>\(0.375\)</td><td>\(0.375\)</td><td>\(0.125\)</td></tr> </table> b) \(P(X_8\le3)=\frac{93}{256}\approx0.3633\), or \(36.33\%\).

- Use the formula for exactly \(k\) successes. - Identify the failure probability. - Recall the general expansion of \((a+b)^n\). - Connect the sum of the expansion terms to the required sum of a probability distribution.

1. Here, \(n=5\), \(p=\frac{1}{3}\), and \(1-p=\frac{2}{3}\). 2. Using \(P(X=k)=\binom{5}{k}\left(\frac{1}{3}\right)^k\left(\frac{2}{3}\right)^{5-k}\), the probabilities are \(\frac{32}{243}\), \(\frac{80}{243}\), \(\frac{80}{243}\), \(\frac{40}{243}\), \(\frac{10}{243}\), and \(\frac{1}{243}\) for \(k=0\) through \(5\), respectively. 3. The binomial expansion \((p+q)^n=\sum_{k=0}^{n}\binom{n}{k}p^kq^{n-k}\) contains exactly the probability terms \(P(X=k)\). Since \(p+q=1\), \(\left(\frac{1}{3}+\frac{2}{3}\right)^5=1\), matching the fact that all probabilities in the distribution add to \(1\).

a) \(P(X=0)=\frac{32}{243}\); \(P(X=1)=\frac{80}{243}\); \(P(X=2)=\frac{80}{243}\); \(P(X=3)=\frac{40}{243}\); \(P(X=4)=\frac{10}{243}\); \(P(X=5)=\frac{1}{243}\). b) The terms in the expansion of \(\left(\frac{1}{3}+\frac{2}{3}\right)^5\) are the probabilities \(P(X=k)\). Their sum is \(1\).

- Find the probability of guessing one question correctly. - Translate “at least two” into the possible values of \(X\). - Decide whether direct addition or a complement is more efficient. - Identify the factor in the binomial formula that counts arrangements.

1. The number of correct answers is binomial with \(n=4\), \(p=\frac{1}{5}=0.2\), and \(1-p=0.8\). 2. The distribution is \(P(X=0)=0.4096\), \(P(X=1)=0.4096\), \(P(X=2)=0.1536\), \(P(X=3)=0.0256\), and \(P(X=4)=0.0016\). 3. \(P(X\ge2)=P(X=2)+P(X=3)+P(X=4)=0.1808\). 4. The coefficient is \(\binom{4}{2}=6\).

a) <table> <tr><th>\(k\)</th><th>\(0\)</th><th>\(1\)</th><th>\(2\)</th><th>\(3\)</th><th>\(4\)</th></tr> <tr><th>\(P(X=k)\)</th><td>\(0.4096\)</td><td>\(0.4096\)</td><td>\(0.1536\)</td><td>\(0.0256\)</td><td>\(0.0016\)</td></tr> </table> b) \(P(X\ge2)=0.1808\). c) \(\binom{4}{2}=6\).

- First find the success probability for one five-shot series. - Use that series-level probability in a second binomial model. - Preserve an exact expression for the very small zero-success probability.

1. For one series, \(p_S=\binom{5}{4}(0.8)^4(0.2)+(0.8)^5=0.73728\). 2. Let \(Y\sim\operatorname{Bin}(12,0.73728)\). Then \(P(Y=8)=\binom{12}{8}(0.73728)^8(0.26272)^4\approx0.2059\). 3. \(P(Y=0)=(0.26272)^{12}\approx1.08\times10^{-7}\).

1. \(0.73728\). 2. \(0.2059\). 3. \((0.26272)^{12}\approx1.08\times10^{-7}\).

- First find the probability that a ten-chip package contains at least one defect. - Treat each package as one Bernoulli trial in the second-stage model. - Keep the package-level probability unrounded in later calculations.

1. A package is accepted only if all ten chips are nondefective, so \(p_R=1-(0.95)^{10}\approx0.4013\). 2. Let \(Y\) be the number of rejected packages. Then \(Y\sim\operatorname{Bin}(20,p_R)\). 3. Using the unrounded value of \(p_R\), \(P(Y=5)=\binom{20}{5}p_R^5(1-p_R)^{15}\approx0.0735\). 4. \(P(Y\le1)=(1-p_R)^{20}+20p_R(1-p_R)^{19}\approx0.000505\).

a) \(p_R\approx0.4013\). b) \(0.0735\). c) \(0.000505\).

- Translate each condition on owners into a condition on nonowners. - Remember that \(X+Y=50\). - Track the inequality direction carefully when converting between \(X\) and \(Y\).

1. Since \(Y=50-X\), \(P(X=42)=P(Y=8)\approx0.1169.\) 2. \(P(X\ge38)=P(Y\le12)\approx0.8139.\) 3. \(P(X\le44)=P(Y\ge6)=1-P(Y\le5)\approx0.9520.\)

(1) \(0.1169\). (2) \(0.8139\). (3) \(0.9520\).

- Translate strict inequalities to inclusive integer endpoints. - Use a difference of cumulative probabilities for the interval. - Use a complement for the upper tail.

1. \(X\sim\operatorname{Bin}(100,0.75)\). 2. \(P(X<70)=P(X\le69)\approx0.1038.\) 3. \(P(72\le X\le80)=P(X\le80)-P(X\le71)\approx0.6929.\) 4. \(P(X>80)=1-P(X\le80)\approx0.0995.\)

(1) \(0.1038\). (2) \(0.6929\). (3) \(0.0995\).

- Use cumulative probabilities for the tail and interval events. - Translate the number with landlines into the complementary count without landlines. - Check whether part b) describes a new event or the same event in different words.

1. \(X\sim\operatorname{Bin}(100,0.12)\). 2. \(P(X\le10)\approx0.3337\). 3. \(P(X>15)=1-P(X\le15)\approx0.1414\). 4. \(P(8\le X\le14)=P(X\le14)-P(X\le7)\approx0.7079\). 5. At least \(90\) households with a landline is equivalent to at most \(10\) without a landline, so the probability is again \(0.3337\).

a) (1) \(0.3337\). (2) \(0.1414\). (3) \(0.7079\). b) \(0.3337\); it is the same event as at most \(10\) households without a landline.

- Find the success probability for one draw. - Use the binomial probability formula for an exact number of successes. - In part b), note that the number of trials changes. - Compare fractions using a common denominator.

1. The success probability is \(p=\frac{1}{3}\), and the failure probability is \(\frac{2}{3}\). 2. \(P(E_1)=\binom{3}{1}\left(\frac{1}{3}\right)\left(\frac{2}{3}\right)^2=\frac{4}{9}\). 3. \(P(E_2)=\binom{3}{2}\left(\frac{1}{3}\right)^2\left(\frac{2}{3}\right)=\frac{2}{9}\). Therefore, \(P(E_1)=2P(E_2)\). 4. \(P(E_3)=\binom{6}{2}\left(\frac{1}{3}\right)^2\left(\frac{2}{3}\right)^4=\frac{80}{243}\). 5. Since \(P(E_1)=\frac{4}{9}=\frac{108}{243}\), it is not twice \(P(E_3)\). In fact, \(\frac{P(E_1)}{P(E_3)}=\frac{27}{20}=1.35\).

a) \(P(E_1)=\frac{4}{9}\) and \(P(E_2)=\frac{2}{9}\), so \(P(E_1)=2P(E_2)\). b) No. \(P(E_1)=\frac{108}{243}\) and \(P(E_3)=\frac{80}{243}\), so \(P(E_1)\ne2P(E_3)\).

- Use the binomial probability formula for the exact-count events. - For “at least one,” use the complement of no wins. - An expected value is an average, not a guaranteed outcome.

1. The success probability is \(p=\frac{1}{4}\). 2. \(P(A)=\binom{4}{1}\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^3=\frac{27}{64}=0.421875\approx0.4219\). 3. \(P(B)=\binom{8}{2}\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)^6=\frac{5103}{16384}\approx0.3115\). The probabilities are not equal. 4. Expected value is a long-run average, not a guarantee for every group of four spins. 5. \(P(C)=1-\left(\frac{3}{4}\right)^4=\frac{175}{256}=0.68359375\approx0.6836<1\), so \(C\) is not certain.

a) \(P(A)=\frac{27}{64}\approx0.4219\) and \(P(B)=\frac{5103}{16384}\approx0.3115\); they are not equal. b) The claim is false. \(P(C)=\frac{175}{256}\approx0.6836\), not \(1\).

- A complement contains exactly the outcomes not in the original event. - Track whether each boundary value is included. - Use cumulative probabilities after writing the complement correctly.

1. \(X\sim\operatorname{Bin}(25,0.45)\). 2. The complement of \(E_1\) is “at most \(10\) made shots,” so \(P(\overline{E_1})=P(X\le10)\approx0.3843.\) 3. The complement of \(E_2\) is “at least \(15\) made shots,” so \(P(\overline{E_2})=P(X\ge15)=1-P(X\le14)\approx0.0960.\)

(1) At most \(10\) made shots; \(0.3843\). (2) At least \(15\) made shots; \(0.0960\).

- Write each original event as an inequality before finding its complement. - The complement of a bounded interval consists of two separated ranges. - Event and complement probabilities add to \(1\).

1. \(X\sim\operatorname{Bin}(50,0.20)\). 2. The complement of \(E_1\) is \(X\le11\), meaning at most eleven people support the statement. Thus, \(P(E_1^c)\approx0.7107\). 3. Event \(E_2\) is \(6\le X\le12\). Its complement is \(X\le5\) or \(X\ge13\). 4. \(P(E_2)=P(X\le12)-P(X\le5)\), so \(P(E_2^c)=1-P(E_2)\approx0.2341\).

a) At most \(11\) people support the statement; \(P(E_1^c)\approx0.7107\). b) At most \(5\) or at least \(13\) people support the statement; \(P(E_2^c)\approx0.2341\).

- Use arrival probability \(0.90\), not the no-show probability, as \(p\). - In part b), use the complement relationship from part a). - Use a point probability for exactly eighty arrivals and a CDF difference for the interval.

1. Each guest arrives with probability \(0.90\), so \(X\sim\operatorname{Bin}(85,0.90)\). 2. \(P(X\le80)\approx0.9357\). 3. The event \(X>80\) is the complement of \(X\le80\), so \(P(X>80)=1-P(X\le80)\approx0.0643\). 4. \(P(X=80)=\binom{85}{80}(0.90)^{80}(0.10)^5\approx0.0717\). 5. \(P(70\le X\le75)=P(X\le75)-P(X\le69)\approx0.3338\).

a) \(0.9357\). b) \(0.0643\); the two events are complements. c) \(0.0717\). d) \(0.3338\).

- Use cumulative probabilities for the tail and interval events. - Find \(np\) before translating the event in part d). - Use the binomial point formula once the mean count is identified.

1. \(X\sim\operatorname{Bin}(200,0.04)\). 2. \(P(X\le5)\approx0.1856\). 3. \(P(X>10)=1-P(X\le10)\approx0.1800\). 4. \(P(6\le X\le12)=P(X\le12)-P(X\le5)\approx0.7545\). 5. The mean is \(\mu=np=8\), so the final event is \(X=8\). Thus, \(P(X=8)=\binom{200}{8}(0.04)^8(0.96)^{192}\approx0.1425\).

a) \(0.1856\). b) \(0.1800\). c) \(0.7545\). d) \(0.1425\).

- State all four conditions for a binomial setting, including the fixed number of trials. - Define success in this context. - Use the binomial formula for an exact number of successes.

1. A binomial model requires a fixed number of trials, two relevant outcomes on each trial, independent trials, and a constant success probability. 2. Here, the fixed number is \(20\) packs. Each pack either contains a limited-edition card or does not. Because the production run is very large, the success probability is approximately constant at \(\frac{1}{7}\), and the selections can be treated as approximately independent. 3. Thus, \(n=20\) and \(p=\frac{1}{7}\). 4. \(P(X=3)=\binom{20}{3}\left(\frac{1}{7}\right)^3\left(\frac{6}{7}\right)^{17}\approx0.2418\).

a) Fixed \(n=20\), two relevant outcomes, approximately independent selections, and approximately constant success probability \(p=\frac{1}{7}\). b) \(n=20\) and \(p=\frac{1}{7}\). c) \(\binom{20}{3}\left(\frac{1}{7}\right)^3\left(\frac{6}{7}\right)^{17}\approx0.2418\).

- Use the binomial probability formula for each value of \(k\). - Organize the values and probabilities in a table. - The expected value is the weighted average of all possible values. - Multiply each value \(k\) by its probability and add.

1. Using \(P(X=k)=\binom{5}{k}(0.4)^k(0.6)^{5-k}\), the probabilities are: \(P(X=0)=0.07776\), \(P(X=1)=0.2592\), \(P(X=2)=0.3456\), \(P(X=3)=0.2304\), \(P(X=4)=0.0768\), and \(P(X=5)=0.01024\). 2. The expected value from the distribution is \(E(X)=0(0.07776)+1(0.2592)+2(0.3456)+3(0.2304)+4(0.0768)+5(0.01024)=2\).

a) <table border="1"> <tr><td>\(k\)</td><td>\(0\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td><td>\(4\)</td><td>\(5\)</td></tr> <tr><td>\(P(X=k)\)</td><td>\(0.07776\)</td><td>\(0.2592\)</td><td>\(0.3456\)</td><td>\(0.2304\)</td><td>\(0.0768\)</td><td>\(0.01024\)</td></tr> </table> b) \(E(X)=2\)

- Use the expected-value formula for a binomial random variable. - Apply the point-probability formula at the mean and its neighboring values. - A mode must have probability at least as large as the probabilities at neighboring values.

1. The success probability is \(p=0.25\), so \(\mu=np=40(0.25)=10.\) 2. \(P(X=10)=\binom{40}{10}(0.25)^{10}(0.75)^{30}\approx0.1444.\) 3. \(P(X=9)=\binom{40}{9}(0.25)^9(0.75)^{31}\approx0.1397,\) and \(P(X=11)=\binom{40}{11}(0.25)^{11}(0.75)^{29}\approx0.1312.\) 4. Since \(0.1444\) exceeds both neighboring probabilities, \(10\) is the most likely value.

a) \(\mu=10\). b) \(P(X=10)\approx0.1444\). c) \(P(X=9)\approx0.1397\) and \(P(X=11)\approx0.1312\); both are less than \(P(X=10)\).

- Find \(\mu\) and \(q\) before forming the stated interval. - Identify the integer contained in that interval. - Compare the candidate mode with its immediate neighbors using binomial point probabilities.

1. The expected value is \(\mu=np=15(0.70)=10.5,\) and \(q=1-0.70=0.30\). 2. The stated interval is \([\mu-q,\mu+p]=[10.2,11.2].\) Its only integer is \(k=11\). 3. \(P(X=10)\approx0.2061,\qquad P(X=11)\approx0.2186,\qquad P(X=12)\approx0.1700.\) 4. Since \(P(X=11)\) is greater than both neighboring probabilities, \(11\) is the mode.

a) \(\mu=10.5\) and \(q=0.30\). b) \([10.2,11.2]\); \(k=11\). c) \(P(X=10)\approx0.2061\), \(P(X=11)\approx0.2186\), and \(P(X=12)\approx0.1700\). Thus, \(11\) is the most likely value.

- Use the same binomial model for all one-day probabilities. - Which counts belong in the event “more than three”? - For the yearly expectation, think about how an expected count across many days relates to the chance on one day.

1. \(X\sim\operatorname{Bin}(300,0.008)\). 2. \(P(X=0)=(0.992)^{300}\approx0.0898\), \(P(X=1)=300(0.008)(0.992)^{299}\approx0.2174\), and \(P(X=2)=\binom{300}{2}(0.008)^2(0.992)^{298}\approx0.2621\). 3. \(P(X>3)=1-P(X\le3)\approx0.2208\). 4. Let \(I_d\) indicate that day \(d\)'s sample has no defective chips. Then \(E(I_d)=P(X=0)\). 5. By linearity of expectation, the expected number of such days is \(365(0.992)^{300}\approx32.79\).

a) \(P(X=0)\approx0.0898\), \(P(X=1)\approx0.2174\), and \(P(X=2)\approx0.2621\). b) \(P(X>3)\approx0.2208\). c) Approximately \(32.79\) days.

- Identify \(n\) and \(p\). - Use the binomial formula for zero and one success. - For at least two successes, subtract the zero- and one-success cases from \(1\).

1. Let \(X\sim\operatorname{Bin}(200,\frac{1}{200})\). 2. \(P(X=0)=\left(\frac{199}{200}\right)^{200}\approx0.3670\). This is about \(0.0009\) less than \(\frac{1}{e}\approx0.3679\). 3. \(P(X=1)=200\left(\frac{1}{200}\right)\left(\frac{199}{200}\right)^{199}=\left(\frac{199}{200}\right)^{199}\approx0.3688\). 4. \(P(X\ge2)=1-P(X=0)-P(X=1)\approx0.2642\).

a) \(P(X=0)\approx0.3670\), about \(0.0009\) less than \(\frac{1}{e}\). b) \(P(X=1)\approx0.3688\). c) \(P(X\ge2)\approx0.2642\).

- Use the supplied probability that one patient is present at the selected time. - Overcrowding means more than ten simultaneous patients. - Compare the calculated probability with the stated one-percent criterion.

1. Each patient is in the waiting room with probability \(p=15/300=0.05\), and the simultaneous waiting-room indicators are assumed independent. Therefore, \(X\sim\operatorname{Bin}(80,0.05)\). 2. Overcrowding occurs when \(X>10\). 3. \(P(X>10)=1-P(X\le10)\approx0.0021\). 4. Since \(0.0021=0.21\%<1\%\), ten seats meet the stated requirement.

The overcrowding probability is approximately \(0.0021\), or \(0.21\%\). Therefore, \(10\) seats are sufficient under the stated criterion.

- Use the binomial formulas for mean and standard deviation. - Check both expected successes and expected failures. - Identify the integer values inside the continuous \(1.96\sigma\) interval. - Compare the exact binomial probability with the normal benchmark.

1. Since \(X\sim\operatorname{Bin}(400,0.15)\), \(\mu=60\) and \(\sigma=\sqrt{51}\approx7.14\). The large-count condition is satisfied because \(np=60\) and \(n(1-p)=340\). 2. The radius is \(1.96\sigma\approx13.997\), so the integer values inside the interval are \(47\) through \(73\). 3. Using the binomial distribution, \(P(47\le X\le73)\approx0.9417\). 4. The exact binomial probability, about \(94.17\%\), is close to the central normal value of \(95\%\), but not identical because the binomial distribution is discrete.

1. \(\mu=60\), \(\sigma=\sqrt{51}\approx7.14\); the normal approximation is appropriate. 2. \(P(47\le X\le73)\approx0.9417\). 3. \(94.17\%\) is close to \(95\%\).

- Ask how much a sample of ten changes a very large population. - “At most one” includes zero and one. - Treat the binomial model here as an approximation, not an exact without-replacement model.

1. Removing only \(10\) chips from a very large shipment changes the remaining defect proportion negligibly. The selections can therefore be treated as approximately independent with approximately constant defect probability \(p=0.04\). 2. Use \(X\approx\operatorname{Bin}(10,0.04)\). 3. \(P(X=1)=\binom{10}{1}(0.04)(0.96)^9\approx0.2770.\) 4. \(P(X\le1)=(0.96)^{10}+10(0.04)(0.96)^9\approx0.9418.\)

a) The sample is negligible compared with the shipment, so the defect probability is approximately constant and the selections are approximately independent. b) \(0.2770\). c) \(0.9418\).

- Write the formula for a binomial point probability. - Compare its factors with the terms in the binomial theorem. - Substitute the success and failure probabilities for \(a\) and \(b\). - Recall the required total probability of a complete distribution.

1. The sum of all point probabilities is \(\sum_{k=0}^{n}P(X=k)=\sum_{k=0}^{n}\binom{n}{k}p^k(1-p)^{n-k}\). 2. In the binomial theorem, set \(a=p\) and \(b=1-p\). 3. The sum becomes \((p+(1-p))^n=1^n=1\). Therefore, every binomial distribution is normalized.

a) \(\sum_{k=0}^{n}P(X=k)=\sum_{k=0}^{n}\binom{n}{k}p^k(1-p)^{n-k}\). b) Substituting \(a=p\) and \(b=1-p\) into the binomial theorem gives \((p+(1-p))^n=1\).

- Compare each summand with the binomial probability formula. - Use \((a+b)^n\) with \(a=0.25\) and \(b=0.75\). - If the zero-success term is omitted, subtract exactly that term from the complete probability sum.

1. Each summand has the form \(\binom{n}{k}p^k(1-p)^{n-k},\) so \(n=12\) and \(p=0.25\). 2. Because the sum includes every possible number of successes, \(T\) is the total probability of the complete binomial distribution. 3. By the binomial theorem, \(T=(0.25+0.75)^{12}=1.\) 4. Starting at \(k=1\) omits only the \(k=0\) term, \((0.75)^{12}\). 5. Therefore, the new value is \(1-(0.75)^{12}.\)

a) \(n=12\), \(p=0.25\), and \(T\) is the sum of all probabilities in the binomial distribution. b) \(T=1\). c) \(1-(0.75)^{12}\).

- Count failures instead of successes. - Relate the two counts by their fixed total of \(20\). - Convert the resulting upper-tail event to a complement if useful.

1. Let \(Y\) be the number of seeds that do not germinate. Then \(Y\sim\operatorname{Bin}(20,0.20).\) 2. The event \(X\le14\) is equivalent to \(Y\ge6\). 3. Therefore, \(P(X\le14)=P(Y\ge6)=1-P(Y\le5)\approx0.1958.\)

\(P(X\le14)\approx0.1958\).

- Translate the interval for correct answers into one for incorrect answers. - Remember that increasing \(X\) decreases \(Y=25-X\). - Use two cumulative probabilities after finding the transformed integer interval.

1. A cumulative distribution lets an inclusive interval probability be found by subtracting two cumulative probabilities instead of adding many individual point probabilities. 2. Since \(Y=25-X\), the condition \(15\le X\le20\) is equivalent to \(5\le Y\le10.\) 3. Therefore, \(P(15\le X\le20)=P(5\le Y\le10) =P(Y\le10)-P(Y\le4) \approx0.7566.\)

a) An inclusive interval can be found as \(P(X\le b)-P(X\le a-1)\), avoiding a long sum of point probabilities. b) \(P(15\le X\le20)\approx0.7566\).

- Replacement makes the one-draw probability stay at \(1/25\). - Use the binomial point formula for zero, one, and two draws of the specified ball. - Use the complement of at most two for part b), and \(np\) for part c).

1. Let \(X\) be the number of times the specified ball is drawn. Replacement makes the draws independent with constant probability \(p=1/25=0.04\), so \(X\sim\operatorname{Bin}(25,0.04)\). 2. \(P(X=0)=(0.96)^{25}\approx0.3604\). 3. \(P(X=1)=\binom{25}{1}(0.04)(0.96)^{24}\approx0.3754\). 4. \(P(X=2)=\binom{25}{2}(0.04)^2(0.96)^{23}\approx0.1877\). 5. \(P(X>2)=1-P(X\le2)\approx0.0765\). 6. \(E(X)=np=25(0.04)=1\).

a) \(0.3604,\ 0.3754,\ 0.1877\), respectively. b) \(0.0765\). c) \(1\).

- Find how many sectors are green. - State all four conditions for a binomial setting. - How many success positions are possible when exactly two spins are green? - Which outcomes belong to “at most one”?

1. The experiment has a fixed \(6\) spins, each with two relevant outcomes (green or not green), independent spins, and a constant chance of green. 2. The spinner has one green section out of five, so \(p=\frac{1}{5}=0.20\) and \(X\sim\operatorname{Bin}(6,0.20)\). 3. \(P(X=2)=\binom{6}{2}(0.20)^2(0.80)^4=0.24576\). 4. \(P(X\le1)=(0.80)^6+6(0.20)(0.80)^5=0.65536\).

a) Fixed \(n=6\), green versus not green, independent spins, and constant \(p=\frac{1}{5}=0.20\); thus \(X\sim\operatorname{Bin}(6,0.20)\). b) \(0.24576\). c) \(0.65536\).

- Read the green and red sector counts from the spinner. - Treat the named color as success in a separate binomial model for each part. - Use a complement for “at least \(2\).”

1. The spinner has \(3\) green sectors out of \(6\), so \(p_G=\frac{1}{2}\). For \(X\sim\operatorname{Bin}(8,\frac{1}{2})\), \(P(X\le3)=\frac{\binom{8}{0}+\binom{8}{1}+\binom{8}{2}+\binom{8}{3}}{2^8}=\frac{93}{256}\approx36.33\%\). 2. The spinner has one red sector out of six, so \(p_R=\frac{1}{6}\). For \(Y\sim\operatorname{Bin}(8,\frac{1}{6})\), \(P(Y\ge2)=1-P(Y\le1)=\frac{663991}{1679616}\approx39.53\%\).

a) \(\frac{93}{256}\approx36.33\%\). b) \(\frac{663991}{1679616}\approx39.53\%\).

- First determine the probability of landing on each requested color from the spinner. - Translate “at most” and “at least ... and at most” into inequalities for the count. - Use the binomial cumulative distribution function. - Express an inclusive interval probability as a difference of two cumulative probabilities.

1. The spinner has \(3\) red sectors and \(5\) blue sectors out of \(10\), so \(p_{\text{red}}=\frac{3}{10}=0.3\) and \(p_{\text{blue}}=\frac{5}{10}=0.5\). 2. Let \(R\) count red results. Then \(R\sim\operatorname{Bin}(30,0.3)\), so \(P(R\le8)\approx0.4315=43.15\%\). 3. Let \(B\) count blue results. Then \(B\sim\operatorname{Bin}(30,0.5)\). 4. \(P(12\le B\le18)=P(B\le18)-P(B\le11)\approx0.7995=79.95\%\).

a) About \(43.15\%\). b) About \(79.95\%\).

- Which features of the repeated spins let you use a binomial model for red? - Read the red and blue probabilities from the wheel. - In part c), which sequences of blue and non-blue results have the required neighboring pair? - Does the usual binomial count distinguish neighboring results from separated ones?

1. A fixed number of independent spins is made. Each result is either red or not red, and the chance of red stays the same on each spin. 2. The wheel has \(2\) red sections out of \(8\), so \(X\sim\operatorname{Bin}(10,0.25)\). 3. \(P(X\le2)=\sum_{k=0}^{2}\binom{10}{k}(0.25)^k(0.75)^{10-k}\approx0.5256\). 4. Blue also occurs with probability \(\frac{1}{4}\). Write B for blue and N for not blue. The valid four-spin patterns are BBNN, NBBN, and NNBB. 5. Each pattern has probability \(\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)^2\), so the required probability is \(3\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)^2=\frac{27}{256}\).

a) The number of spins is fixed, the spins are independent, and the chance of red remains constant. Every spin is either red or not red. Thus \(n=10\), \(p=\frac{1}{4}\), and \(X\sim\operatorname{Bin}(10,0.25)\). b) \(P(X\le2)\approx0.5256\). c) \(\frac{27}{256}\).

- Find the chance of yellow and not yellow on one spin. - How many different three-spin sequences contain two yellow results? - Think about what one product of three probabilities counts. - Compare whether each expression represents one sequence or all the sequences required.

1. Three of the eight sectors are yellow, giving \(P(\text{yellow})=\frac{3}{8}\) and \(P(\text{not yellow})=\frac{5}{8}\). 2. Exactly two yellow outcomes can occur in three orders: YY-N, Y-NY, and N-YY, where N means not yellow. 3. Each order has probability \(\left(\frac{3}{8}\right)^2\frac{5}{8}\). 4. Expression (3) adds the three orders, while (4) multiplies one order's probability by \(3\). Both are correct. 5. Expression (1) counts only one order. Expression (2) instead counts exactly one yellow and two non-yellow results.

Expressions (3) and (4) are correct: (3) adds all three possible orders and (4) counts them by multiplying one order's probability by \(3\). Expression (1) counts only one order; (2) counts exactly one yellow result.

- Match \(0.30\) with purple and \(0.70\) with orange. - Compare the first expression with a binomial point probability. - A leading \(1-\) indicates a complement. - Read the summation limits as the included values of \(X\).

1. The probability of purple is \(0.30\), and the probability of orange is \(0.70\). 2. Expression 1 is \(P(X=2)\): purple occurs exactly twice. 3. Expression 2 is \(P(X=0)\): every result is orange. 4. Expression 3 is \(1-P(X=0)=P(X\ge1)\): purple occurs at least once. 5. Expression 4 is \(P(X\le2)\): purple occurs at most twice.

1) Purple occurs exactly \(2\) times. 2) Purple never occurs; every result is orange. 3) Purple occurs at least once. 4) Purple occurs at most \(2\) times.

- Read the chance of each named color from the spinner. - Does “exactly” describe one count or several? What about “at most”? - In part b), which color has the one-spin probability appearing in the expression?

1. From the wheel, \(P(\text{green})=\frac{3}{8}\), \(P(\text{red})=\frac{1}{4}\), and \(P(\text{yellow})=\frac{1}{8}=0.125\). 2. For event A, \(P(A)=\binom{50}{20}\left(\frac{3}{8}\right)^{20}\left(\frac{5}{8}\right)^{30}\approx0.107230\), or \(10.72\%\). 3. For event B, \(P(B)=\sum_{k=0}^{10}\binom{50}{k}\left(\frac{1}{4}\right)^k\left(\frac{3}{4}\right)^{50-k}\approx0.262202\), or \(26.22\%\). 4. The \(0.125\) in part b) is the probability of yellow on one spin. The sum counts \(0\) through \(8\) yellow results in \(60\) independent spins.

a) Event A: \(10.72\%\). Event B: \(26.22\%\). b) In \(60\) independent spins, yellow appears at most \(8\) times.

- Use the same exact-count binomial formula under each proposed value of \(p\). - Compare the two probabilities for the observed count. - Distinguish “more plausible under these models” from “proved correct.”

1. Under Model A, \(P(X=17)=\binom{20}{17}(0.95)^{17}(0.05)^3\approx0.05958.\) 2. Under Model B, \(P(X=17)=\binom{20}{17}(0.80)^{17}(0.20)^3\approx0.20536.\) 3. The observed count has the larger point probability under Model B, so Model B makes this particular observation more plausible. 4. This comparison does not prove that Model B is true; it compares only how likely this observed count is under the two specified models.

a) Model A: \(0.05958\); Model B: \(0.20536\). b) Model B makes exactly \(17\) germinations more plausible, but the comparison does not prove that Model B is correct.

- Identify the success and failure probabilities for one guessed question. - Use the binomial formula for each possible value of \(X\). - Compare the binomial coefficients with the number of arrangements having the same number of correct answers.

1. Here, \(X\sim\operatorname{Bin}(3,0.25)\). 2. \(P(X=0)=0.75^3=0.421875\), \(P(X=1)=3(0.25)(0.75)^2=0.421875\), \(P(X=2)=3(0.25)^2(0.75)=0.140625\), and \(P(X=3)=0.25^3=0.015625\). 3. In \((p+q)^3=p^3+3p^2q+3pq^2+q^3\), the four terms give the probabilities of \(3,2,1,0\) correct answers, respectively. 4. The binomial coefficients count the possible orders of correct and incorrect answers with each success count.

a) <table> <tr><th>\(k\)</th><th>\(0\)</th><th>\(1\)</th><th>\(2\)</th><th>\(3\)</th></tr> <tr><th>\(P(X=k)\)</th><td>\(0.421875\)</td><td>\(0.421875\)</td><td>\(0.140625\)</td><td>\(0.015625\)</td></tr> </table> b) The terms \(0.25^3\), \(3(0.25)^2(0.75)\), \(3(0.25)(0.75)^2\), and \(0.75^3\) are \(P(X=3)\), \(P(X=2)\), \(P(X=1)\), and \(P(X=0)\), respectively.

- Identify how many separate binomial probabilities are being added. - Use the exponents to determine the number of trials. - Use the bases of the powers to identify the success and failure probabilities. - A sum of several consecutive cases often corresponds to wording such as “at least” or “at most.”

1. The three terms are binomial probabilities for \(n=4\) trials with success probability \(p=\frac{2}{3}\). 2. The first term is \(P(X=4)\), the second is \(P(X=3)\), and the third is \(P(X=2)\). 3. Therefore, the sum represents \(P(X\ge 2)\). 4. One matching experiment is to place \(2\) red marbles and \(1\) blue marble in a bag, draw a marble \(4\) times with replacement, and count a red marble as a success. Event \(B\) is drawing a red marble at least \(2\) times.

One possible experiment is to draw a marble \(4\) times with replacement from a bag containing \(2\) red marbles and \(1\) blue marble. Event \(B\): A red marble is drawn at least \(2\) times.

- At each spin, count how many outcomes would make the new partial sum a multiple of \(5\). - Use the resulting one-game probability as the success probability for the repeated games. - Apply the binomial formula for exactly five successful games.

1. At each stage exactly one of the five possible next values would make the current partial sum divisible by \(5\). 2. Therefore, \(p=P(\text{no partial sum divisible by }5)=\left(\frac45\right)^3=\frac{64}{125}=0.512.\) 3. Let \(K\sim\operatorname{Bin}(10,64/125)\). Then \(P(K=5)=\binom{10}{5}\left(\frac{64}{125}\right)^5\left(\frac{61}{125}\right)^5\approx0.2454.\)

a) \(\frac{64}{125}=0.512\). b) \(\binom{10}{5}(64/125)^5(61/125)^5\approx0.2454\).

- A probability is larger than the preceding one when their ratio is greater than \(1\). - Form the ratio of two consecutive binomial probabilities. - Interpret what happens when the ratio equals \(1\). - Simplify the ratio of the consecutive binomial coefficients.

1. For a binomial random variable, \(\frac{P(X=k)}{P(X=k-1)}=\frac{n-k+1}{k}\frac{p}{1-p}\). 2. Substituting \(n=14\) and \(p=0.4\) gives \(\frac{P(X=k)}{P(X=k-1)}=\frac{15-k}{k}\frac{2}{3}\). 3. The probabilities are nondecreasing while this ratio is at least \(1\): \(\frac{15-k}{k}\frac{2}{3}\ge1\). 4. Solving gives \(2(15-k)\ge3k\), so \(30\ge5k\), or \(k\le6\). 5. At \(k=6\), the ratio equals \(1\), so \(P(X=6)=P(X=5)\). Before that point the probabilities increase, and after that point they decrease. 6. Therefore, the maximum occurs at both \(k=5\) and \(k=6\).

\(P(X=k)\) is greatest for \(k=5\) and \(k=6\).

- Find the probability of one correct guess first. - Translate conditions on incorrect answers into conditions on the number correct when useful. - In part c), separate the first three fixed successes from the remaining seventeen questions. - In part d), treat the first five and last five questions as independent blocks.

1. The probability of a correct answer is \(p=0.25\), and the probability of an incorrect answer is \(0.75\). 2. Part a) specifies one fixed sequence, so \(P=(0.25)^5(0.75)^{15}=1.305025489\ldots\times10^{-5}\approx1.305\times10^{-5}.\) 3. More than \(15\) incorrect answers means fewer than \(5\) correct answers. If \(X\) is the total number correct, \(P(X\le4)\approx0.4148.\) 4. In part c), the first \(3\) answers are already correct, so exactly \(3\) of the remaining \(17\) must be correct: \(P=\binom{17}{3}(0.25)^6(0.75)^{14}\approx0.0030.\) 5. For part d), let \(Y_1\) be the number correct among the first \(5\), and let \(Y_2\) be the number correct among the last \(5\). 6. \(P(Y_1\ge4)=\binom54(0.25)^4(0.75)+(0.25)^5=0.015625\). 7. \(P(Y_2\le1)=(0.75)^5+5(0.25)(0.75)^4=0.6328125\). 8. The two blocks are independent, so \(P(Y_1\ge4,\ Y_2\le1)=0.015625(0.6328125)\approx0.0099.\)

a) \(P\approx1.305\times10^{-5}\). b) \(P\approx0.4148\). c) \(P\approx0.0030\). d) \(P\approx0.0099\).

- Substitute \(k=0\), \(k=1\), and \(k=2\) into the binomial formula. - Use exponent rules to simplify ratios with the same base. - Expand the binomial coefficient for two successes. - Evaluate the limiting value of each factor separately.

1. \(P(X=0)=\left(1-\frac{1}{n}\right)^n\), while \(P(X=1)=n\left(\frac{1}{n}\right)\left(1-\frac{1}{n}\right)^{n-1}=\left(1-\frac{1}{n}\right)^{n-1}\). 2. Therefore, \(\frac{P(X=1)}{P(X=0)}=\frac{1}{1-\frac{1}{n}}=\frac{n}{n-1}\), which proves part a). 3. For two successes, \(P(X=2)=\binom{n}{2}\left(\frac{1}{n}\right)^2\left(1-\frac{1}{n}\right)^{n-2}\). 4. Simplifying gives \(P(X=2)=\frac{n-1}{2n}\left(1-\frac{1}{n}\right)^{n-2}\). 5. As \(n\) becomes large, \(\frac{n-1}{2n}\to\frac{1}{2}\), and \(\left(1-\frac{1}{n}\right)^{n-2}\to\frac{1}{e}\). Therefore, \(P(X=2)\approx\frac{1}{2e}\).

a) \(\frac{P(X=1)}{P(X=0)}=\frac{n}{n-1}\), so \(P(X=1)=\frac{n}{n-1}P(X=0)\). b) \(P(X=2)=\frac{n-1}{2n}\left(1-\frac{1}{n}\right)^{n-2}\approx\frac{1}{2}\cdot\frac{1}{e}=\frac{1}{2e}\) for large \(n\).

- Compare the variance formulas after interchanging \(p\) and \(1-p\). - Relate the red and blue counts using their fixed total. - Transform the interval for \(X\) into the corresponding interval for \(Y\).

1. For \(X\), \(\sigma_X=\sqrt{192(0.25)(0.75)}=6\). For \(Y\), \(\sigma_Y=\sqrt{192(0.75)(0.25)}=6\). 2. The mean of \(X\) is \(\mu_X=192(0.25)=48\), so the two-standard-deviation interval is \([36,60]\). 3. \(P(36\le X\le60)=P(X\le60)-P(X\le35)\approx0.9632\). 4. Since \(Y=192-X\), the event \(36\le X\le60\) is equivalent to \(132\le Y\le156\). 5. Because \(\mu_Y=144\) and \(\sigma_Y=6\), \([132, 156]\) is exactly the two-standard-deviation interval for \(Y\). Thus, the two probabilities are equal.

1. \(\sigma_X=6\) and \(\sigma_Y=6\). 2. \(P(36\le X\le60)\approx0.9632\). 3. The probabilities are equal because \(Y=192-X\), so the two intervals describe the same outcomes.