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Experimental probability and simulations

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5171117
Two bags contain gold and silver marbles. Bag A contains \(10\) gold marbles and \(90\) silver marbles. Bag B contains \(50\) gold marbles and \(50\) silver marbles. A student repeatedly drew a marble from one bag, recorded its color, and replaced it. In \(10\) draws, the student drew gold \(4\) times and silver \(6\) times. Which bag was the student more likely using? Explain by comparing the theoretical probabilities with the experimental result.

Hints

- Find the theoretical probability of gold for each bag. - Write the experimental number of gold draws as a fraction of all draws. - Compare the experimental fraction with the two theoretical probabilities.

Solution

1. In Bag A, the theoretical probability of gold is \(\frac{10}{100} = \frac{1}{10}\). 2. In Bag B, the theoretical probability of gold is \(\frac{50}{100} = \frac{1}{2}\). 3. The experimental relative frequency of gold is \(\frac{4}{10} = \frac{2}{5}\). 4. The value \(\frac{2}{5}\) is much closer to \(\frac{1}{2}\) than to \(\frac{1}{10}\), so the results are more consistent with Bag B. This does not prove which bag was used, but Bag B is more likely.

Answer

Bag B is more likely. Its gold-marble probability is \(\frac{1}{2}\), and the experimental result \(\frac{4}{10} = \frac{2}{5}\) is much closer to \(\frac{1}{2}\) than to Bag A’s \(\frac{1}{10}\).
5171127
Two containers hold different mixtures of blue and white buttons. Container 1: \(30\) blue buttons and \(10\) white buttons Container 2: \(10\) blue buttons and \(30\) white buttons In each experiment, a button was drawn, recorded, and replaced four times. Experiment A: \(1\) blue and \(3\) white Experiment B: \(3\) blue and \(1\) white Which experiment is more likely to have used each container? Explain.

Hints

- Find the theoretical fraction of blue buttons in each container. - Find the experimental fraction of blue draws in each experiment. - Match equal or closest fractions.

Solution

1. In Container 1, the theoretical probability of blue is \(\frac{30}{40} = \frac{3}{4}\). Experiment B has an experimental blue frequency of \(\frac{3}{4}\), so it matches Container 1 more closely. 2. In Container 2, the theoretical probability of blue is \(\frac{10}{40} = \frac{1}{4}\). Experiment A has an experimental blue frequency of \(\frac{1}{4}\), so it matches Container 2 more closely. 3. Therefore, the more likely assignment is Container 1 to Experiment B and Container 2 to Experiment A.

Answer

Container 1 most likely produced Experiment B, and Container 2 most likely produced Experiment A.
5271577
Match each statement with the type of probability it uses, and briefly explain your choice. Statement A: “The prize wheel has \(20\) equal sections, and only one section awards the grand prize, so the probability of winning is \(\frac{1}{20}\).” Statement B: “I tossed this irregular building block \(500\) times. It landed on its narrow side \(142\) times, so I estimate the probability of that result as \(28.4\%\).” Types: - Theoretical probability - Experimental probability

Hints

- Was the probability determined before or after repeated trials? - Are equally likely outcomes being counted? - Did the value come from a mathematical model or from observed data?

Solution

1. Statement A uses theoretical probability because it is based on equally likely outcomes: \(P=\frac{1}{20}\). 2. Statement B uses experimental probability because the estimate is the observed relative frequency \(\frac{142}{500}=0.284\).

Answer

A: Theoretical probability, because the sections are equally likely. B: Experimental probability, because the estimate comes from observed relative frequency.
5271697
For each example, identify the probability approach being used and briefly explain your choice. a) The probability of rolling a prime number on a fair eight-sided die is stated as \(0.5\). b) Statistical data report that the probability of a multiple birth is about \(1.2\%\). c) A meteorologist gives a \(30\%\) chance of rain based on weather data from the past \(50\) years. Approaches: theoretical probability or experimental probability.

Hints

- Ask whether equally likely outcomes are being counted. - Look for values based on observations or historical data. - Decide whether each value comes from a mathematical model or collected data.

Solution

1. Part a) uses theoretical probability. A fair die has equally likely faces, and the prime labels \(2,3,5,7\) give \(P=\frac{4}{8}=0.5\). 2. Part b) uses experimental probability because the value comes from observed statistical data. 3. Part c) uses an experimental probability approach because the estimate is based on past observations.

Answer

a) Theoretical probability, based on equally likely die faces. b) Experimental probability, based on statistical data. c) Experimental probability, based on past weather observations.
5271717
For each statement, identify whether it uses theoretical or experimental probability. Briefly explain your choice. a) Demographic data give the probability that a newborn is male as \(51.4\%\). b) A spinner has \(10\) equal sections labeled \(1\) through \(10\). The probability of landing on a prime number is \(0.4\). c) Insurance records give a \(0.02\%\) probability of a wildlife collision on a particular road per trip.

Hints

- Determine whether the value comes from counting equally likely outcomes. - Look for values based on past observations. - Ask whether physical symmetry makes the outcomes equally likely.

Solution

1. Part a) is experimental probability because the value is based on birth records. 2. Part b) is theoretical probability because the equal sections are equally likely and \(4\) of the \(10\) labels are prime. 3. Part c) is experimental probability because the value comes from past insurance data.

Answer

a) Experimental probability b) Theoretical probability c) Experimental probability
5271767
A computer program randomly arranges the four letters \(A\), \(B\), \(C\), and \(D\), using each letter exactly once. a) Assuming all arrangements are equally likely, find the theoretical probability that \(A\) is first. b) In \(10{,}000\) runs, \(A\) appeared first \(2540\) times. Find the relative frequency and identify the type of probability represented by this value.

Hints

- Compare the one favorable first letter with the four possible first letters. - Relative frequency is occurrences divided by trials. - Distinguish a calculated probability from one estimated using data.

Solution

1. Each of the four letters is equally likely to be first, and only one of them is \(A\). Therefore, \(P(A\text{ first})=\frac{1}{4}=0.25\). This is theoretical probability. 2. The relative frequency is \(\frac{2540}{10000}=0.254\). This is experimental probability because it is based on observed results.

Answer

a) \(0.25\); theoretical probability b) \(0.254\); experimental probability
5271897
Researchers report that a new treatment has a probability of success of \(p=0.75\). 1. Explain whether this value would ordinarily come from an equally likely-outcomes model or from experimental data. 2. Explain why an equally likely-outcomes model is not appropriate for determining this probability. 3. Describe how the relative frequency of successful treatments in a study of \(n\) participants is related to \(p\) as \(n\) becomes large.

Hints

- Ask whether the probability can be found by counting symmetric outcomes. - Consider how medical probabilities are usually estimated. - Recall what happens to relative frequency over many trials.

Solution

1. The value would ordinarily be estimated from experimental data collected in clinical studies, so it represents experimental probability. 2. There is no symmetry or counting argument that makes treatment success and failure equally likely. Biological outcomes must be studied rather than assigned equal probabilities. 3. By the law of large numbers, under consistent conditions the relative frequency of successful treatments tends to stabilize near the underlying probability \(p=0.75\) as the sample size increases.

Answer

1. The value is ordinarily based on experimental data. 2. Success and failure are not known to be equally likely, so an equally likely-outcomes model is not justified. 3. For large \(n\), the relative frequency of success tends to approach \(0.75\).
5308157
An irregularly shaped object can come to rest in positions \(A\), \(B\), or \(C\). The table shows cumulative results as the number of tosses increases. <table> <thead> <tr> <th>Number of tosses \(n\)</th> <th>Position \(A\)</th> <th>Position \(B\)</th> <th>Position \(C\)</th> </tr> </thead> <tbody> <tr> <td>\(200\)</td> <td>\(82\)</td> <td>\(74\)</td> <td>\(44\)</td> </tr> <tr> <td>\(400\)</td> <td>\(170\)</td> <td>\(142\)</td> <td>\(88\)</td> </tr> <tr> <td>\(600\)</td> <td>\(250\)</td> <td>\(218\)</td> <td>\(132\)</td> </tr> <tr> <td>\(800\)</td> <td>\(336\)</td> <td>\(288\)</td> <td>\(176\)</td> </tr> </tbody> </table> a) Find the relative frequency of each position after \(800\) tosses. b) Based on these data, estimate how many times each position would occur in \(2400\) tosses.

Hints

- Divide each count by the total number of tosses. - Use the largest available trial count for the most stable estimate. - Multiply each estimated probability by \(2400\).

Solution

1. After \(800\) tosses, the relative frequencies are \(\frac{336}{800}=0.42\) for position \(A\), \(\frac{288}{800}=0.36\) for position \(B\), and \(\frac{176}{800}=0.22\) for position \(C\). 2. Using these relative frequencies as estimates, the predicted counts in \(2400\) tosses are \(0.42\cdot2400=1008\) for \(A\), \(0.36\cdot2400=864\) for \(B\), and \(0.22\cdot2400=528\) for \(C\).

Answer

a) Position \(A\): \(0.42\); position \(B\): \(0.36\); position \(C\): \(0.22\) b) Approximately \(1008\) times in position \(A\), \(864\) times in position \(B\), and \(528\) times in position \(C\)
5308167
A weighted two-sided game token is tossed \(1000\) times. It lands star side up \(372\) times and circle side up \(628\) times. a) Explain why an equally likely-outcomes model is not suitable for predicting these probabilities before the experiment. b) Estimate the probability that the token lands circle side up. c) Based on the experiment, estimate how many star-side-up results would occur in \(5000\) tosses.

Hints

- Consider whether the device has physical symmetry. - Use relative frequency to estimate probability. - Multiply the estimated star-side-up probability by \(5000\).

Solution

1. The token is weighted, so there is no reason to assume that its two sides are equally likely to land face up. 2. The estimated probability of landing circle side up is \(\frac{628}{1000}=0.628\). 3. The relative frequency of star-side-up results is \(\frac{372}{1000}=0.372\). The estimated number in \(5000\) tosses is \(0.372\cdot5000=1860\).

Answer

a) The token is weighted, so the two outcomes need not be equally likely. b) \(0.628\) c) Approximately \(1860\) star-side-up results
5308277
A custom deck contains \(16\) cards: the hearts and diamonds with ranks \(7,8,9,10,J,Q,K,A\). One card is drawn at random. a) Assuming all cards are equally likely, find the theoretical probability of drawing an ace. b) In \(400\) draws with replacement, an ace was drawn \(48\) times. Find the relative frequency and compare it with the theoretical probability. c) Does the experimental result provide evidence against the equally likely-card model? Explain using the relationship between relative frequency and probability.

Hints

- Count the aces and the total cards. - Divide the observed ace count by \(400\). - A relative frequency does not have to equal the theoretical probability exactly.

Solution

1. The deck has \(16\) cards and \(2\) aces, so \(P(\text{ace})=\frac{2}{16}=0.125\). 2. The relative frequency is \(\frac{48}{400}=0.12\), which is \(0.005\) below the theoretical probability. 3. The result does not provide meaningful evidence against the model. Random variation can produce a relative frequency slightly different from the theoretical probability, and \(0.12\) is close to \(0.125\).

Answer

a) \(0.125\) b) \(0.12\), which is \(0.005\) below the theoretical value c) No. The small difference is consistent with ordinary random variation.
5309267
A spinner has \(10\) equal sections labeled with the digits \(0\) through \(9\). It is spun twice. 1. Find the theoretical probability that the two results have a sum of exactly \(10\). 2. Find the probability that the product of the two results is odd. 3. In \(10{,}000\) computer-simulated trials, a sum of \(10\) occurred \(924\) times. Find the relative frequency and compare it with the theoretical probability.

Hints

- Use ordered pairs for the two spins. - A product is odd only when both factors are odd. - Relative frequency is successes divided by trials.

Solution

1. There are \(10\cdot10=100\) equally likely ordered pairs. The pairs with sum \(10\) are \((1, 9),(2, 8),(3, 7),(4, 6),(5, 5)\) and their reverses, for \(9\) outcomes. Thus, \(P=\frac{9}{100}=0.09\). 2. A product is odd only if both digits are odd. There are \(5\) odd digits, so \(P=\frac{5}{10}\cdot\frac{5}{10}=0.25\). 3. The relative frequency is \(\frac{924}{10000}=0.0924\), which is \(0.0024\) above the theoretical probability \(0.09\).

Answer

1. \(0.09\) 2. \(0.25\) 3. \(0.0924\), which is close to \(0.09\)
5309327
A fair twelve-sided die is rolled three times. a) Find the theoretical probability that at least one result is repeated. b) In \(1000\) simulated trials of three rolls, at least one repeat occurred in \(241\) trials. Find the relative frequency and its absolute difference from the theoretical probability. c) Explain the relationship between the simulation result and the theoretical probability using the law of large numbers.

Hints

- Use the complement that all three results are different. - Divide simulated successes by \(1000\). - Finite simulations can differ from the theoretical value.

Solution

1. The complement is that all three results are different. Thus, \(P(\text{all different})=\frac{12\cdot11\cdot10}{12^3}=\frac{55}{72}\). Therefore, \(P(\text{at least one repeat})=1-\frac{55}{72}=\frac{17}{72}\approx 0.23611\). 2. The relative frequency is \(\frac{241}{1000}=0.241\). The absolute difference is \(\left|0.241-\frac{17}{72}\right|\approx 0.00489\). 3. As the number of independent simulation trials increases, the relative frequency tends to stabilize near the theoretical probability, though finite simulations still vary.

Answer

a) \(\frac{17}{72}\approx 0.23611\) b) Relative frequency \(0.241\); absolute difference approximately \(0.00489\) c) With more trials, the relative frequency tends to approach the theoretical probability.
5375077
Two coin simulations produced the frequencies shown. Which simulation gives more convincing support for the claim that the coin is fair? Justify your answer using relative frequencies.
Figure for problem 537507

Hints

- Do not compare only the absolute counts. - Find the proportion of heads in each simulation.

Solution

1. Simulation A: The relative frequency of heads is \(\frac{31}{50} = 0.62\). 2. Simulation B: The relative frequency of heads is \(\frac{2511}{5000} = 0.5022\). 3. The distances from \(0.5\) are \(0.12\) for A and \(0.0022\) for B. 4. Simulation B gives more convincing support because its relative frequency is much closer to \(0.5\) and is based on many more trials.

Answer

Simulation B; its relative frequency of heads, \(0.5022\), is much closer to \(0.5\).
5375417
An event has a theoretical probability of \(0.3\). In \(1000\) simulated trials, the event occurred \(287\) times. Is this outcome reasonable for the probability model? Explain.
Figure for problem 537541

Hints

- Calculate the experimental probability from the simulation results. - Decide whether a finite simulation must match the theoretical probability exactly.

Solution

1. The experimental probability is \(\frac{287}{1000}=0.287\). 2. Its difference from the theoretical probability is \(0.3-0.287=0.013\). 3. Simulation results vary, and \(0.287\) is close to \(0.3\). The outcome is reasonable for the model.

Answer

Yes. The experimental probability \(0.287\) is close to the theoretical probability \(0.3\).
5126917
A computer places \(500\) points uniformly at random inside a square. The square has side length \(40\,\text{cm}\), and the largest possible circle is drawn inside it. Of the \(500\) points, \(392\) land inside the circle. Use the simulation results to estimate \(\pi\).

Hints

- How is the circle's radius related to the square's side length? - What fraction of the square's area is inside the circle? - How can the proportion of random points inside the circle estimate the area ratio? - How can you solve the area-ratio equation for \(\pi\)?

Solution

1. The experimental proportion of points inside the circle is \(\frac{392}{500} = 0.784\). 2. If the square has side length \(s\), then the circle has radius \(\frac{s}{2}\). The area ratio is \(\frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{\pi(\frac{s}{2})^2}{s^2} = \frac{\pi}{4}\). 3. Use the experimental proportion to estimate the area ratio: \(\frac{\pi}{4} \approx 0.784\). 4. Multiply by \(4\): \(\pi \approx 4 \cdot 0.784 = 3.136\).

Answer

\(\pi \approx 3.136\)
5135567
Three student groups estimate how often the letter S appears in English text. - Group A examines \(50\) letters and finds S \(2\) times. - Group B examines \(500\) letters and finds S \(36\) times. - Group C combines the class results: S appears \(310\) times among \(5000\) letters. a) Find the relative frequency of S for each group. b) Which value is probably the best estimate of the long-run proportion of S in similar English texts? Justify your choice using a statistical principle.

Hints

- Compare the number of letters examined by each group. - Think about what usually happens to relative frequency as the sample size grows. - A larger sample is less affected by a few unusual observations.

Solution

1. a) Group A: \(\frac{2}{50} = 0.04 = 4\%\). Group B: \(\frac{36}{500} = 0.072 = 7.2\%\). Group C: \(\frac{310}{5000} = 0.062 = 6.2\%\). 2. b) Group C provides the best estimate because it uses the largest sample. By the law of large numbers, relative frequencies generally become more stable as the number of observations increases. A larger sample does not guarantee an exact value, but it usually reduces the effect of random variation.

Answer

a) Group A: \(4\%\); Group B: \(7.2\%\); Group C: \(6.2\%\) b) Group C, because its sample is largest and its relative frequency is generally the most stable estimate.
5135577
A computer program counts vowels in a passage containing exactly \(250\) letters: A: \(15\), E: \(32\), I: \(20\), O: \(8\), U: \(12\). a) Find the relative frequency of vowels in the passage. b) A student says, “If a reference model gives the letter E a probability of \(12.7\%\), then every \(250\)-letter English passage must contain E exactly \(31.75\) times.” Evaluate the claim using both the calculation and the meaning of a probability model.

Hints

- Add the five vowel counts first. - Can a passage contain a fractional number of letters? - Does a long-run probability require every individual sample to match it exactly?

Solution

1. a) The total number of vowels is \(15 + 32 + 20 + 8 + 12 = 87\). 2. The relative frequency of vowels is \(\frac{87}{250} = 0.348 = 34.8\%\). 3. b) The model calculation is \(250 \cdot 0.127 = 31.75\). This is an expected or long-run average count, not a required count in every passage. 4. A letter count must be a whole number, and individual passages vary because of their vocabulary and subject. Therefore, the student’s claim is false.

Answer

a) \(34.8\%\) b) The claim is false. The value \(31.75\) is a model-based expected count, not a count that every passage must have, and actual letter counts must be whole numbers.
5135647
A computer simulates drawing one ball from a bag containing ten balls numbered \(1\) through \(10\). The ball is replaced after every draw. The table shows how often the number \(7\) was drawn in different numbers of trials. <table> <tr><td>Number of trials \(n\)</td><td>\(100\)</td><td>\(1000\)</td><td>\(10{,}000\)</td><td>\(100{,}000\)</td></tr> <tr><td>Number of 7s</td><td>\(13\)</td><td>\(92\)</td><td>\(1045\)</td><td>\(9982\)</td></tr> </table> a) Find the relative frequency of drawing a \(7\) for each set of trials. b) Find the theoretical probability of drawing a \(7\). c) Compare the relative frequencies with the theoretical probability. Explain how the results illustrate the law of large numbers.

Hints

- Divide the number of 7s by the total number of trials. - All ten numbered balls are equally likely. - Compare each experimental result with \(0.1\).

Solution

1. a) For \(n = 100\), the relative frequency is \(\frac{13}{100} = 0.13\). For \(n = 1000\), the relative frequency is \(\frac{92}{1000} = 0.092\). For \(n = 10{,}000\), the relative frequency is \(\frac{1045}{10000} = 0.1045\). For \(n = 100{,}000\), the relative frequency is \(\frac{9982}{100000} = 0.09982\). 2. b) The ten outcomes are equally likely, so \(P(7) = \frac{1}{10} = 0.1\). 3. c) The absolute differences from \(0.1\) are \(0.03\), \(0.008\), \(0.0045\), and \(0.00018\). In these trial sets, the relative frequency gets closer to \(0.1\) as the number of trials increases. This illustrates the law of large numbers: over many repetitions, experimental probability tends to stabilize near theoretical probability.

Answer

a) \(0.13\), \(0.092\), \(0.1045\), and \(0.09982\) b) \(0.1 = 10\%\) c) As the number of trials increases, the relative frequency shown becomes closer to \(0.1\), illustrating the law of large numbers.
5135657
Two seventh-grade classes investigate whether a coin is fair. Class 7A flips the coin \(250\) times and gets tails \(140\) times. Class 7B flips the coin \(2500\) times and gets tails \(1280\) times. a) Find the relative frequency of tails for each class. b) For a fair coin, the theoretical probability of tails is \(50\%\). Use the law of large numbers to explain which result gives more reliable evidence about whether the coin is fair.

Hints

- Compare the number of flips completed by the two classes. - Consider how sample size affects random variation. - Decide which relative frequency is the more stable estimate of \(0.5\).

Solution

1. a) Class 7A: \(\frac{140}{250} = 0.56\). Class 7B: \(\frac{1280}{2500} = 0.512\). 2. b) Class 7B used ten times as many flips. By the law of large numbers, relative frequency generally becomes more stable around theoretical probability as the number of trials increases. Therefore, the value \(0.512\) from Class 7B is the more reliable estimate. The law does not require the results to move closer to \(0.5\) after every additional flip.

Answer

a) Class 7A: \(0.56 = 56\%\); Class 7B: \(0.512 = 51.2\%\) b) Class 7B gives more reliable evidence because it used many more trials, so random variation has less influence on the relative frequency.
5271587
A weighted six-sided die is being tested. Two students discuss the probability of rolling a \(6\). Luke says, “The die has six sides, so every number must have probability \(\frac{1}{6}\).” Sarah says, “I rolled the die \(200\) times and got a \(6\) on \(52\) rolls, so I estimate the probability at about \(26\%\).” 1. Explain the assumption behind Luke’s reasoning and why it may be invalid here. 2. Find the relative frequency Sarah used and name the principle supporting her estimate. 3. Describe how the probability could be estimated more accurately.

Hints

- Ask whether the physical device makes all outcomes equally likely. - Relative frequency is successes divided by trials. - Consider how increasing the number of trials affects random variation.

Solution

1. Luke assumes that all six outcomes are equally likely. Because the die is described as weighted, that symmetry assumption may be false. 2. Sarah’s relative frequency is \(\frac{52}{200}=0.26\). The law of large numbers supports using a long-run relative frequency to estimate an unknown probability. 3. Roll the die many more times under the same conditions. A larger number of trials generally produces a more stable relative-frequency estimate.

Answer

1. Luke assumes equally likely outcomes, which is not justified for a weighted die. 2. The relative frequency is \(0.26\), or \(26\%\); the supporting principle is the law of large numbers. 3. Conduct many more rolls under consistent conditions and recompute the relative frequency.
5271607
A bottle cap is tossed \(500\) times. It lands open side up \(320\) times and top side up \(180\) times. a) Explain why assuming the two outcomes are equally likely is not appropriate. b) Use the experimental results to estimate the probability that the bottle cap lands open side up. c) Explain how relative frequency is connected to probability as the number of tosses becomes very large.

Hints

- Consider whether the object has physical symmetry. - Divide the number of open-side-up results by the total number of tosses. - Think about how random variation changes with many trials.

Solution

1. A bottle cap is physically asymmetric, so there is no reason to assume that its two resting positions are equally likely. 2. The experimental estimate is \(\frac{320}{500}=0.64\). 3. Under repeated independent trials in the same conditions, the relative frequency tends to stabilize near the underlying probability as the number of trials grows. This is the law of large numbers.

Answer

a) A bottle cap is not symmetric, so the two outcomes need not be equally likely. b) \(0.64\), or \(64\%\) c) With many repeated trials, the relative frequency tends to approach the underlying probability.
5271707
A student investigates how a thumbtack lands when tossed: on its round head, event \(H\), or on its side, event \(S\). a) Explain why equally likely outcomes cannot be assumed to predict an exact value of \(P(H)\). b) In \(500\) tosses, the thumbtack lands on its head \(185\) times. Use experimental probability to estimate \(P(H)\). c) State the conditions that \(P(H)\) and \(P(S)\) must satisfy to form a valid two-outcome probability model.

Hints

- Equally likely outcomes require an appropriate symmetry argument. - Experimental probability is an observed relative frequency. - Probabilities for all outcomes in the sample space must be nonnegative and sum to \(1\).

Solution

1. The thumbtack is not physically symmetric, so there is no reason to assume that the two outcomes are equally likely. 2. The experimental estimate is \(P(H)\approx\frac{185}{500}=0.37\). 3. A valid model requires \(P(H)\ge 0\), \(P(S)\ge 0\), and \(P(H)+P(S)=1\).

Answer

a) The outcomes cannot be assumed equally likely because the thumbtack is not symmetric. b) \(P(H)\approx 0.37\). c) \(P(H),P(S)\ge 0\) and \(P(H)+P(S)=1\).
5271727
An urn contains \(4\) red balls and \(6\) blue balls that differ only in color. a) Use theoretical probability to find the probability of drawing a red ball. b) In \(50\) trials with replacement, a red ball is drawn \(18\) times. Find the relative frequency and identify the associated probability approach. c) Explain why the result in part b) can differ from the theoretical probability in part a).

Hints

- For equally likely balls, divide favorable outcomes by total outcomes. - Relative frequency is successes divided by trials. - Consider the effect of random variation in a limited sample.

Solution

1. The theoretical probability is \(P(\text{red})=\frac{4}{10}=0.40\). 2. The relative frequency is \(\frac{18}{50}=0.36\), which is an experimental probability estimate. 3. Relative frequencies vary from sample to sample. With more trials, the relative frequency generally becomes more stable around the theoretical probability.

Answer

a) \(P(\text{red})=0.40\) b) The relative frequency is \(0.36\); this is experimental probability. c) Random variation in a finite number of trials can make the experimental result differ from the theoretical value.
5308217
A large transportation study in a metropolitan area found that \(18\%\) of households have no car, \(54\%\) have exactly one car, and \(28\%\) have two or more cars. a) Describe the random selection process and model assumption needed for the statement “The probability that a randomly selected household has no car is approximately \(0.18\)” to be meaningful. b) In the context of the law of large numbers, distinguish the observed relative frequency \(0.18\) from the probability in the underlying model.

Hints

- Identify what makes the household sample representative in the probability model. - Distinguish an observed sample proportion from the probability it estimates. - Recall what the law of large numbers says about repeated random trials.

Solution

1. The experiment is to select one household at random from the metropolitan-area population. The sampling method should give every household the same chance of selection. Under that model, the observed proportion \(0.18\) can be used to estimate the probability that a selected household has no car. 2. The relative frequency \(0.18\) is a value observed in one sample. The model probability is the long-run probability for the population process. Across many independent random selections under the same conditions, relative frequencies tend to stabilize near that probability, but one study’s value need not equal it exactly.

Answer

a) Select a household randomly so that every household has the same chance of selection; then \(0.18\) is an estimate of the model probability. b) The relative frequency is an observed sample value, while the model probability is the long-run value that relative frequencies tend to approach.
5308227
A container holds an unknown number of red, blue, and green balls. In \(500\) draws with replacement, a red ball was drawn \(124\) times. a) Find the relative frequency of drawing a red ball and explain why it can be used to estimate the probability \(p\). b) A student says, “There are three colors, so each color must have probability \(\frac{1}{3}\).” Evaluate this claim using the experimental result.

Hints

- Relative frequency is occurrences divided by trials. - Ask what information would make the three colors equally likely. - Compare the experimental result with \(\frac{1}{3}\).

Solution

1. The relative frequency is \(\frac{124}{500}=0.248\). With many repeated draws under the same conditions, relative frequency tends to stabilize near the underlying probability, so \(0.248\) is a reasonable estimate of \(p\). 2. The value \(\frac{1}{3}\) would require the three colors to be equally likely, such as having equal numbers of balls of each color. That information is not given. The observed value \(0.248\) also differs substantially from \(\frac{1}{3}\), so the equal-probability claim is not justified.

Answer

a) \(0.248\); it is an experimental estimate of \(p\). b) The claim is not justified because three possible colors do not automatically make the colors equally likely, and the observed relative frequency is well below \(\frac{1}{3}\).
5308317
A fair eight-sided die is labeled \(1\) through \(8\). 1. Let \(A\) be the event that the result is prime or a perfect square. List the outcomes in \(A\) and find \(P(A)\). 2. In \(250\) rolls, a prime occurred \(120\) times and a perfect square occurred \(65\) times. No result is both prime and a perfect square, so event \(A\) occurred \(185\) times. Find the relative frequency of \(A\) and compare it with the theoretical probability. 3. Explain how the relative frequency is expected to relate to the theoretical probability as the number of rolls increases.

Hints

- List the primes and perfect squares from \(1\) through \(8\). - Divide the number of occurrences of \(A\) by \(250\). - Recall what happens to relative frequency over many trials.

Solution

1. The primes are \(2,3,5,7\), and the perfect squares are \(1,4\). Thus, \(A=\{1,2,3,4,5,7\}\), so \(P(A)=\frac{6}{8}=0.75\). 2. The relative frequency is \(\frac{185}{250}=0.74\), which is \(0.01\) below the theoretical probability. 3. By the law of large numbers, the relative frequency tends to become more stable near the theoretical probability as the number of rolls increases.

Answer

1. \(A=\{1,2,3,4,5,7\}\) and \(P(A)=0.75\) 2. The relative frequency is \(0.74\), which is close to \(0.75\). 3. With more rolls, the relative frequency tends to approach the theoretical probability.
5308587
A spinner has \(10\) equal sections labeled \(1\) through \(10\). Let \(E\) be the event that the result is prime or odd. a) Find the theoretical probability \(P(E)\). b) In \(200\) spins, event \(E\) occurred \(132\) times. Find the relative frequency and compare it with the theoretical probability. c) Does this result provide strong evidence that the spinner is not fair? Explain.

Hints

- Form the union of the prime and odd outcomes without double-counting. - Divide the number of occurrences by \(200\). - Remember that finite experiments fluctuate.

Solution

1. The primes are \(2,3,5,7\), and the odd numbers are \(1,3,5,7,9\). Their union is \(\{1,2,3,5,7,9\}\), so \(P(E)=\frac{6}{10}=0.6\). 2. The relative frequency is \(\frac{132}{200}=0.66\), which is \(0.06\) above the theoretical probability. 3. A finite experiment can differ from its theoretical probability because of random variation. This result alone is not strong enough to conclude that the spinner is unfair.

Answer

a) \(0.6\) b) \(0.66\), which is \(0.06\) higher than the theoretical probability c) No. The difference could result from ordinary random variation.
5309377
A six-sided die is rolled \(1200\) times, and a \(6\) occurs \(246\) times. a) Find the relative frequency of rolling a \(6\) and compare it with the theoretical probability for a fair die. b) Using the law of large numbers, evaluate whether the die appears to be fair. c) Suppose the observed relative frequency is used as the actual probability \(p\) of rolling a \(6\). Find the probability of rolling at least one \(6\) in three independent rolls.

Hints

- Compare the observed relative frequency with the fair-die probability. - A large sample should usually produce a relative frequency near the true probability. - Use the complement of no \(6\) in three rolls.

Solution

1. The relative frequency is \(\frac{246}{1200}=0.205\). A fair die would give \(P(6)=\frac{1}{6}\approx 0.1667\). 2. The observed relative frequency is about \(0.0383\) above the fair-die probability. With \(1200\) trials, the law of large numbers suggests that the relative frequency should usually be fairly stable near the true probability. This sizable gap suggests that the die may not be fair, although additional testing would provide stronger evidence. 3. Using \(p=0.205\), \(P(\text{at least one }6)=1-(1-0.205)^3=1-(0.795)^3=0.497555125\).

Answer

a) Relative frequency \(0.205\); fair-die probability approximately \(0.1667\) b) The result suggests that the die may be weighted, although more testing would strengthen the conclusion. c) \(0.497555125\approx 0.4976\)
5375947
Use random digits \(0\) through \(9\) to simulate two independent trials with success probability \(0.3\). Give one possible assignment of digits to “success” and “failure,” and find the probability of at least one success.
Figure for problem 537594

Hints

- Assign exactly \(3\) of the \(10\) equally likely digits to success. - Use the complement of two failures.

Solution

1. One possible assignment is success for \(\{0, 1, 2\}\) and failure for \(\{3, 4, 5, 6, 7, 8, 9\}\). 2. The probability of no success in two trials is \(0.7^2 = 0.49\). 3. Therefore, \(P(\text{at least one success}) = 1 - 0.49 = 0.51\).

Answer

For example, assign \(0\), \(1\), and \(2\) to success; \(P(\text{at least one success}) = 0.51\).
5135667
A factory checks parts made by three machines. Each machine is expected to have a defect rate of \(5\%\). The inspection gives these results: - Machine A: \(12\) defective parts out of \(200\) inspected. - Machine B: \(105\) defective parts out of \(2000\) inspected. - Machine C: \(1160\) defective parts out of \(20{,}000\) inspected. a) Find the relative frequency of defective parts for each machine. b) Based on these data, which machine gives the strongest indication that its defect probability differs from \(0.05\)? Explain using the law of large numbers.

Hints

- Divide the number of defective parts by the number inspected. - Compare each relative frequency with \(0.05\). - A deviation based on many trials is generally less likely to be caused only by random variation.

Solution

1. a) Machine A: \(\frac{12}{200} = 0.06\). Machine B: \(\frac{105}{2000} = 0.0525\). Machine C: \(\frac{1160}{20000} = 0.058\). 2. The absolute differences from \(0.05\) are \(0.01\), \(0.0025\), and \(0.008\), respectively. 3. Random variation usually has less effect when many parts are inspected. Machine C still has a relative frequency noticeably above \(0.05\) after \(20{,}000\) inspections, so its data give the strongest indication of a different defect probability. This is evidence, not proof.

Answer

a) Machine A: \(0.06\); Machine B: \(0.0525\); Machine C: \(0.058\) b) Machine C, because its relative frequency remains noticeably above \(0.05\) despite the very large number of inspections.

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