Free math worksheets

A random variable \(X\) is uniformly distributed over \(M=\{10,20,30,40,50,60,70,80,90,100\}\). Find \(E(X)\).

A professional biathlete hits a standing-shooting target with probability \(p=0.85\) on each shot. Assume the \(n=1200\) shots in a training season are independent. a) Find the expected number of hits. b) Explain the meaning of this value in context.

A game has expected winnings of \(\$1\) per play. Does this mean a player should expect to win exactly \(\$1\) on the next play? Explain what the expected value means.

Two random variables have the same mean. Random variable \(A\) has standard deviation \(2\), and random variable \(B\) has standard deviation \(7\). Which random variable has greater spread? Explain using the standard deviations.

An automobile manufacturer finds that \(8\%\) of its vehicles have a minor paint defect. Assume vehicles are independently classified with this constant defect probability. A sample of \(150\) vehicles is inspected. Let \(X\) be the number with a paint defect. Find the mean and standard deviation of \(X\).

A spinner has \(10\) equal sections, and exactly one section is labeled Grand Prize. The spinner is spun independently \(80\) times. Let \(X\) be the number of Grand Prizes. Find the mean and standard deviation of \(X\). Briefly interpret the mean in context.

A binomial random variable \(X\) has parameters \(n\) and \(p\). Suppose the number of trials is multiplied by \(25\) while the success probability remains unchanged. Describe how the mean \(E(X)\) and standard deviation \(\sigma(X)\) change.

A binomial random variable \(X\) has parameters \(n\) and \(p\), where \(n\ge1\). Suppose \(\operatorname{Var}(X)=0\). Find all possible values of \(p\), and briefly explain what each value means for the possible values of \(X\).

A binomial random variable \(X\) has success probability \(p=0.25\) and standard deviation \(\sigma=3\). Find the number of trials \(n\), and then find the mean \(\mu\).

A multiple-choice test has \(60\) questions. Each question has \(4\) answer choices, and exactly one choice is correct. Fatou selects an answer at random for every question. The random variable \(X\) is the number of correct answers. a) Find the mean \(\mu\) of \(X\), and interpret it in context. b) Find the probability that Fatou answers exactly the expected number of questions correctly. Give an exact binomial expression and a decimal approximation to four decimal places.

The probability distribution of a random variable \(X\) is shown below: <table> <tr> <td>\(x_i\)</td> <td>\(10\)</td> <td>\(20\)</td> <td>\(30\)</td> <td>\(40\)</td> </tr> <tr> <td>\(P(X=x_i)\)</td> <td>\(0.4\)</td> <td>\(0.3\)</td> <td>\(0.2\)</td> <td>\(0.1\)</td> </tr> </table> Find the mean \(\mu\), variance \(\operatorname{Var}(X)\), and standard deviation \(\sigma(X)\).

A fair \(12\)-sided die is labeled with the integers \(1\) through \(12\). Let \(X\) be the number rolled. Find the mean \(\mu\) and standard deviation \(\sigma\) of \(X\). Give the standard deviation exactly and to the nearest hundredth.

A fair four-sided die has faces labeled \(1\), \(2\), \(2\), and \(5\). The random variable \(X\) is the number showing after one roll. a) Find \(E(X)\). b) Evaluate this statement: “Because every face is labeled with an integer, the expected value must also be an integer.” Justify your answer using part a).

A raffle drum contains tickets with possible prize values. Let \(X\) be the prize value in dollars on a randomly selected ticket. It is known that \(E(X)=\$5.00\) and \(\sigma(X)=0\). Determine which prize values can appear and find \(P(X=5)\). Explain whether the drum can contain any nonwinning tickets worth \(\$0\).

At a carnival game, a player draws one ball from an urn containing \(20\) balls: \(2\) gold, \(5\) silver, and \(13\) white. A gold ball pays \(\$15.00\), a silver ball pays \(\$5.00\), and a white ball pays nothing. What entry fee makes the game mathematically fair?

A random variable \(Z\) can take the values \(5\) and \(0\), where \(P(Z=5)=p\) and \(P(Z=0)=1-p\). a) Write \(E(Z)\) in terms of \(p\). b) Write \(\operatorname{Var}(Z)\) in terms of \(p\). c) Find \(\sigma(Z)\) when \(p=0.4\).

A binomial random variable \(X\) has mean \(\mu=24\) and standard deviation \(\sigma=4\). Find the parameters \(n\) and \(p\).

Find the parameters \(n\) and \(p\) of a binomial random variable \(X\) for each set of summary statistics. a) \(\mu=10\) and \(\sigma=\sqrt{8}\) b) \(\mu=48\) and \(\sigma=4\)

A long, six-faced die has two square end faces labeled \(1\) and \(2\), and four rectangular side faces labeled \(3\), \(4\), \(5\), and \(6\). Its geometry makes the face probabilities unequal: \(P(1)=P(2)=0.12\) and \(P(3)=P(4)=P(5)=P(6)=0.19\). The random variable \(X\) is the number rolled. a) Find the probability that the number rolled is at least \(4\). b) Find the probability of rolling a prime number. c) Calculate the expected value \(E(X)\).

In a manufacturing process, a sensor has probability \(p\) of having a production-related inaccuracy. Find the expected number of sensors with inaccuracies in each situation. 1. \(p=0.025\) and \(n=1200\) sensors produced in one day 2. \(p=0.008\) and \(n=15{,}000\) sensors in one weekly shipment 3. \(p=0.12\) and \(n=450\) sensors in a test run

The table shows the probability distribution of a binomial random variable \(X\) with \(n=3\). <table border="1"> <tr><td>\(k\)</td><td>\(0\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td></tr> <tr><td>\(P(X=k)\)</td><td>\(0.064\)</td><td>\(0.288\)</td><td>\(0.432\)</td><td>\(0.216\)</td></tr> </table> a) Find \(E(X)\) directly from the table using the definition of expected value for a discrete random variable. b) Use \(E(X)\) to determine the success probability \(p\).

A multiple-choice test has \(10\) questions. Each question has four answer choices, exactly one of which is correct. A test-taker guesses independently and uniformly at random on every question. Let \(X\) be the number of correct answers. a) Find the mean and variance of \(X\). b) Find the standard deviation exactly and to two decimal places, and interpret it in context. c) How does the variance change if the test is expanded to \(20\) questions? Justify your answer using the binomial variance formula.

A binomial random variable \(X\) counts the successes in \(3\) trials with success probability \(p=0.5\). Find the mean \(\mu\), and then calculate the variance directly from the definition \(\operatorname{Var}(X)=\sum_{k=0}^{3}(k-\mu)^2P(X=k)\).

A school raffle ticket costs \(\$2.50\). The payout \(X\) has the following distribution. <table> <tr> <td>Payout \(x_i\)</td> <td>\(\$0\)</td> <td>\(\$5\)</td> <td>\(\$10\)</td> </tr> <tr> <td>Probability \(P(X=x_i)\)</td> <td>\(0.80\)</td> <td>\(0.15\)</td> <td>\(0.05\)</td> </tr> </table> a) Find the expected net gain per ticket for a player. b) Interpret this expected value for a very large number of purchased tickets. c) What ticket price would make the game fair?

A quality-control process finds that \(2.5\%\) of LED lights are defective. a) How many defective lights are expected in a batch of \(600\)? b) After production improvements, only \(1\) in every \(100\) lights is defective. How large must a batch be to have the same expected number of defective lights as in part a)?

A medical study includes \(250{,}000\) participants. The marginal probabilities of four side-effect categories are shown. <table> <tr><th>Side-effect category</th><th>Probability \(p\)</th></tr> <tr><td>None</td><td>\(0.9350\)</td></tr> <tr><td>Mild</td><td>\(0.0520\)</td></tr> <tr><td>Moderate</td><td>\(0.0115\)</td></tr> <tr><td>Severe</td><td>\(0.0015\)</td></tr> </table> Find the expected number of participants in each category. State what assumption about the category probabilities is needed for these expected counts. Then state the additional assumption needed if the full vector of category counts is to be modeled as multinomial.

In an electronics manufacturing process, \(5\%\) of LEDs are defective. Assume LEDs are independently classified with this constant defect probability. A sample of \(20\) LEDs is selected, and \(X\) is the number of defective LEDs. a) Find the probability that at most \(1\) LED is defective. Round to four decimal places. b) Find the mean \(\mu\) and standard deviation \(\sigma\) of \(X\). Give \(\sigma\) exactly and to four decimal places. c) List all possible values of \(X\) in the interval \([\mu-\sigma,\mu+\sigma]\).

A spinner has three equal sections colored red, blue, and yellow. It is spun independently \(12\) times. Let \(Y\) be the number of times it lands on red. a) Find \(P(Y=4)\). Round to four decimal places. b) Find the mean \(\mu\) and standard deviation \(\sigma\) of \(Y\). Give \(\sigma\) exactly and to three decimal places. c) Find the probability that \(Y\) differs from its mean by more than one standard deviation. Round to four decimal places.

A company manufactures glass bottles. Based on past data, \(4\%\) of the bottles have minor material defects. Assume bottles are independently classified with this constant defect probability. A quality-control sample of \(150\) bottles is selected. Let \(X\) be the number of bottles in the sample that have material defects. a) Find the mean \(\mu\) and standard deviation \(\sigma\) of \(X\). b) Find the probability that \(X\) is within one standard deviation of its mean. Round the probability to four decimal places.

In a carnival game, a player wins each round independently with probability \(p=0.25\). The player completes \(80\) rounds. Find the probability that the number of wins differs from its mean by more than two standard deviations. Round the probability to four decimal places.

For a binomial random variable \(X\) with a fixed success probability \(p\), the number of trials \(n\) is changed so that the standard deviation \(\sigma(X)\) becomes four times as large. Determine how the mean \(E(X)\) changes.

Two archers, Alex and Maya, practice hitting a target. Alex hits the target with probability \(p_A=0.60\) and takes \(n_A=80\) shots. Maya hits the target with probability \(p_M=0.75\) and takes \(n_M=64\) shots. Assume each archer's shots are independent and that each hit probability remains constant. Show that their expected numbers of hits are equal. Then find each standard deviation in exact form and as a decimal to two decimal places, and determine whose number of hits varies less around the mean.

A basketball player makes a free throw with probability \(0.75\). During one practice session, the player takes \(40\) free throws. Let \(X\) be the number of made free throws. a) State the conditions under which \(X\) can be modeled by a binomial distribution, and identify \(n\) and \(p\). b) Find the probability that the player makes at least \(35\) free throws. Round to four decimal places. c) Find the mean \(\mu\) and standard deviation \(\sigma\) of \(X\). Give \(\sigma\) exactly and to four decimal places. d) Find the probability that \(X\) lies in the interval \([\mu-\sigma,\mu+\sigma]\). Round to four decimal places.

The random variable \(X\) has a binomial distribution with \(n=10\). The mean \(E(X)\) is known to be an integer. The table shows part of the probability distribution. <table> <tr> <td>\(k\)</td> <td>\(0\)</td> <td>\(1\)</td> <td>\(2\)</td> <td>\(3\)</td> <td>\(4\)</td> </tr> <tr> <td>\(P(X=k)\)</td> <td>\(0.3487\)</td> <td>\(0.3874\)</td> <td>\(0.1937\)</td> <td>\(0.0574\)</td> <td>\(0.0112\)</td> </tr> </table> a) Determine the success probability \(p\). b) Find \(P(1\le X\le3)\). c) Find \(P(X\ge1)\).

The lifetime \(T\), in years, of an electronic component has density \(g(t)=kt^2(3-t)\quad\text{for }0\le t\le3,\) and \(g(t)=0\) otherwise. a) Show that \(g\) is a probability density function when \(k=\frac{4}{27}\). b) Find the mean lifetime \(E(T)\). c) Find the probability that a component lasts longer than two years.

A spinner has four sections numbered \(1\) through \(4\). The probability of landing on a section is proportional to its number, so \(P(X=k)=ck\). The game pays \(\$2.00\) for a \(1\) or \(2\), \(\$5.00\) for a \(3\), and \(\$10.00\) for a \(4\). What entry fee should the game operator charge to earn an average profit of \(\$0.50\) per play over the long run?

In a game, a fair coin is tossed \(5\) times. The random variable \(X\) is the number of heads. The payout in dollars is \(A=X^2\), and the game costs \(\$8.00\) to play. Determine whether the game is favorable to the player in the long run. Find the expected gain or loss per play.

A random variable \(Y\) has the following probability distribution: <table> <tr> <td>\(y_i\)</td> <td>\(-2\)</td> <td>\(-1\)</td> <td>\(0\)</td> <td>\(1\)</td> <td>\(2\)</td> </tr> <tr> <td>\(P(Y=y_i)\)</td> <td>\(0.15\)</td> <td>\(0.25\)</td> <td>\(0.30\)</td> <td>\(0.20\)</td> <td>\(0.10\)</td> </tr> </table> Find the variance \(\operatorname{Var}(Y)\) and standard deviation \(\sigma(Y)\). Round the standard deviation to the nearest hundredth.

A discrete random variable \(X\) can take the values \(-10\), \(0\), \(10\), and \(20\). Three of the probabilities are shown below: <table> <tr> <td>\(x_i\)</td> <td>\(-10\)</td> <td>\(0\)</td> <td>\(10\)</td> <td>\(20\)</td> </tr> <tr> <td>\(P(X=x_i)\)</td> <td>\(0.1\)</td> <td>\(0.5\)</td> <td></td> <td>\(0.1\)</td> </tr> </table> a) Find \(P(X=10)\). b) Find the mean \(E(X)\). c) Find the standard deviation \(\sigma(X)\).

The random variable \(Y\) takes the values \(2\), \(5\), and \(k\) with the probabilities shown below: <table> <tr> <td>\(y_i\)</td> <td>\(2\)</td> <td>\(5\)</td> <td>\(k\)</td> </tr> <tr> <td>\(P(Y=y_i)\)</td> <td>\(0.4\)</td> <td>\(0.4\)</td> <td>\(0.2\)</td> </tr> </table> The mean is \(E(Y)=4.4\). a) Find \(k\). b) Find \(\operatorname{Var}(Y)\).

A bag contains \(6\) balls: \(4\) green balls labeled \(2\) and \(2\) yellow balls labeled \(5\). Two balls are drawn one at a time with replacement. Let \(S\) be the sum of the two numbers drawn. Find \(E(S)\) and \(\sigma(S)\).

One of the six words in the sentence “Statistics is a branch of mathematics” is selected at random. Let \(X\) be the number of letters in the selected word, and let \(Y\) be the number of vowels \((a, e, i, o, u)\) in the word. Find the mean and variance of both random variables.

An inventory system contains five items with identification codes A10, B200, C30, D4000, and E50. One item is selected at random for inspection. Let \(X\) be the number of digits in the selected code, and let \(Y\) be the sum of those digits. Find the mean and variance of \(X\) and \(Y\).

A random variable \(Z\) can take only the values \(0\) and \(5\). Its expected value is \(E(Z)=3.5\). a) Determine the probability distribution of \(Z\), including \(P(Z=0)\) and \(P(Z=5)\). b) Interpret \(3.5\) in terms of a very large number of repetitions of the experiment.

A bakery records the number of muffins purchased by each customer on a Monday morning. Let \(X\) be the number of muffins purchased by a randomly selected customer. <table> <tr><th>\(x\)</th><th>\(0\)</th><th>\(1\)</th><th>\(2\)</th><th>\(3\)</th></tr> <tr><th>\(P(X=x)\)</th><td>\(0.2\)</td><td>\(0.5\)</td><td>\(0.2\)</td><td>\(0.1\)</td></tr> </table> a) Find \(E(X)\) and interpret it in context. b) Find \(\sigma(X)\). Give the exact form and a decimal approximation to four decimal places. c) Find the probability that a randomly selected customer buys more muffins than the mean.

During a grocery-store promotion, each participating customer spins a prize wheel and receives a discount coupon. Assume every coupon is redeemed in full. Let \(Y\) be the coupon value in dollars. <table> <tr><th>\(y\) (dollars)</th><th>\(0\)</th><th>\(2\)</th><th>\(5\)</th><th>\(10\)</th></tr> <tr><th>\(P(Y=y)\)</th><td>\(p\)</td><td>\(0.4\)</td><td>\(0.2\)</td><td>\(0.1\)</td></tr> </table> a) Find \(p\). b) Find \(E(Y)\). What total coupon cost should the store expect for \(500\) participating customers? c) Find \(\operatorname{Var}(Y)\) and \(\sigma(Y)\). Give the standard deviation exactly and to two decimal places.

A random variable \(X\) can take the values \(0\), \(4\), and \(10\). Its probability distribution is partially shown below: <table> <tr> <td>\(x_i\)</td> <td>\(0\)</td> <td>\(4\)</td> <td>\(10\)</td> </tr> <tr> <td>\(P(X=x_i)\)</td> <td>\(p_1\)</td> <td>\(0.3\)</td> <td>\(p_2\)</td> </tr> </table> The mean is \(E(X)=5.2\). a) Find \(p_1\) and \(p_2\). b) Find \(\operatorname{Var}(X)\).

A random variable \(Y\) can take the values \(-2\), \(0\), and \(4\). It is known that \(P(Y=0)=0.4\) and that the mean is \(\mu=-0.6\). a) Find \(P(Y=-2)\) and \(P(Y=4)\). b) Find the standard deviation \(\sigma\) of \(Y\).

A drawer contains \(5\) USB drives. Two contain an important presentation, and the other three are empty. Drives are selected at random and checked one at a time without replacement. The random variable \(X\) is the number of drives checked before the first drive containing the presentation is found. a) Give the probability distribution of \(X\) in a table. b) Find \(E(X)\).

A technician inspects a shipment of \(7\) components, including \(3\) defective components. The components are selected at random and tested one at a time without replacement until the first defective component is found. The random variable \(X\) is the number of tests required. a) What is the greatest possible number of tests? b) What is the probability that a defective component is found on the first or second test? c) Find the expected number of tests.

A discrete random variable \(X\) can take the values \(0\), \(2\), \(4\), and \(6\). Its probability mass function is \(P(X=x)=k(x+1)\), where \(k\) is a positive constant. a) Find \(k\). b) Find \(E(X)\).

The random variable \(Y\) can take the values \(1\), \(2\), \(3\), and \(4\), with \(P(Y=1)=c\), \(P(Y=2)=2c\), \(P(Y=3)=0.3\), and \(P(Y=4)=0.1\). Find \(c\), and then calculate \(E(Y)\).

A random variable \(X\) can take the values \(10\) and \(20\). Let \(P(X=10)=p\), where \(0\le p\le1\). a) Write \(\operatorname{Var}(X)\) as a function of \(p\). b) Find the values of \(p\) for which \(\operatorname{Var}(X)=0\). c) Interpret the results from part b) in terms of the distribution of \(X\).

Eleni claims, “The numerical value of a standard deviation \(\sigma\) can never be greater than the numerical value of its variance \(V\), because \(\sigma=\sqrt{V}\) and taking a square root always makes a number smaller.” a) Evaluate the claim. For \(V>0\), state the condition under which \(\sigma>V\). b) Let \(X\) take the values \(0\) and \(1\), each with probability \(0.5\). Find \(\operatorname{Var}(X)\) and \(\sigma(X)\), and use them as a counterexample.

A spinner is divided into four sections labeled with payouts of \(\$0\), \(\$2\), \(\$5\), and \(\$9\). a) Find the entry fee that makes the game fair when all four sections have equal area. b) In a new version, the entry fee is \(\$3.50\). Give one possible set of central angles for the four sections that keeps the game fair. Explain your reasoning.

A prize box contains tickets worth \(\$0\), \(\$10\), or \(\$50\). a) Initially, the probabilities of drawing the three ticket values are \(70\%\), \(20\%\), and \(10\%\), respectively. Find the fair price of one ticket. b) The box is refilled with exactly \(40\) tickets, and the price of one ticket is set at \(\$10.00\). Give one possible number of tickets at each prize value that makes the game fair. There must be at least one ticket of each value.

A game consists of tossing a fair coin \(3\) times and costs \(\$3.00\) to play. A player receives \(\$x\) for \(3\) heads, \(\$4.00\) for exactly \(2\) heads, and no payout otherwise. Find \(x\) so that the game is fair.

A company analyzes the possible profit from a new project. Let \(X\) be the profit in thousands of dollars. The probability distribution is shown below: <table> <tr><td>Profit \(x_i\)</td><td>\(-5\)</td><td>\(0\)</td><td>\(10\)</td><td>\(20\)</td></tr> <tr><td>\(P(X=x_i)\)</td><td>\(0.2\)</td><td>\(0.3\)</td><td>\(0.4\)</td><td>\(0.1\)</td></tr> </table> Calculate \(S_1=\sum_{i=1}^{4}x_iP(X=x_i)\) and \(S_2=\sum_{i=1}^{4}(x_i-S_1)^2P(X=x_i)\). Interpret both quantities in the context of the project's profit.

A game has three possible payouts: \(\$0\), \(\$5\), and \(\$a\). Their probabilities are \(P(X=0)=0.5\), \(P(X=5)=0.4\), and \(P(X=a)=0.1\). Find \(a\) so that the expected payout is exactly \(\$4\). Then find \(\operatorname{Var}(X)\).

A random variable \(X\) takes the value \(1\) with probability \(p\) and the value \(0\) with probability \(1-p\). Determine the value of \(p\in[0,1]\) for which the standard deviation \(\sigma(X)\) is greatest.

A discrete random variable \(X\) has possible values \(\{2, 4, 6\}\). It is known that \(E(X)=4.2\) and \(\operatorname{Var}(X)=1.16\). Find \(P(X=2)\), \(P(X=4)\), and \(P(X=6)\).

A sensor costs a manufacturer \(\$15\) to produce. Historically, \(94\%\) of the sensors work properly. If a customer receives a defective sensor, the company sends a free replacement. Producing and performing an additional quality check on the replacement costs \(\$18\). What selling price gives the company an expected profit of \(\$4\) per sensor sold?

An online printing company sells photo books for \(\$14.50\). The variable production cost is \(\$8.00\) per book. When a customer files a valid complaint, the company refunds the full purchase price and also incurs \(\$5.00\) in complaint-handling costs. The current complaint rate is \(4\%\). a) Find the company’s expected profit per order. b) What is the greatest complaint rate that keeps the expected profit from falling below \(\$5.00\) per order?

A random variable \(X\) can take the values \(1, 2, 3, 4, 5\). Its probability distribution is symmetric about the mean \(E(X)=3\): <table> <tr> <td>\(k\)</td> <td>\(1\)</td> <td>\(2\)</td> <td>\(3\)</td> <td>\(4\)</td> <td>\(5\)</td> </tr> <tr> <td>\(P(X=k)\)</td> <td>\(p\)</td> <td>\(q\)</td> <td>\(0.2\)</td> <td>\(q\)</td> <td>\(p\)</td> </tr> </table> Given that \(\operatorname{Var}(X)=2.48\), find \(p\) and \(q\).

A discrete random variable \(Z\) takes the values \(-2\), \(0\), and \(2\), where \(P(Z=-2)=a\), \(P(Z=0)=b\), and \(P(Z=2)=c\). Given that \(E(Z)=0.4\) and \(\operatorname{Var}(Z)=2.24\), find \(a\), \(b\), and \(c\).

An electronics store offers a prize game. First, a player draws one ball from a container holding \(25\) balls, of which \(5\) are marked. Only a player who draws a marked ball may spin a prize wheel with three sections: Gold, Silver, and Bronze. Gold awards a \(\$100\) gift card, Silver awards a \(\$40\) gift card, and Bronze awards a \(\$10\) gift card. The probability of Bronze is \(50\%\). The wheel is designed so that the expected prize value per game entry is \(\$7\). Find the probabilities of landing on Gold and Silver.

A bag contains five balls labeled \(1, 1, 2, 3, 3\). Two balls are drawn at the same time without replacement. Let \(X\) be the sum of the two labels. a) Find the probability distribution of \(X\) and display it in a table. b) Find \(E(X)\) and \(\operatorname{Var}(X)\).

In a manufacturing process, \(5\%\) of components are defective. Assume components are independently classified with this constant defect probability. A sample of \(80\) components is selected. Let \(Y\) be the number of defective components. 1. Find the probability that more than \(6\) components are defective. Round to four decimal places. 2. Find the probability that the number of defective components is within one standard deviation of the mean. Include exactly the integer values in the interval and round the probability to four decimal places.

A multiple-choice test has \(30\) questions, each with four answer choices and exactly one correct answer. A test-taker guesses independently and uniformly at random on every question. Let \(X\) be the number of correct answers and \(Y\) the number of incorrect answers. a) Find \(E(X)\), \(E(Y)\), \(\sigma(X)\), and \(\sigma(Y)\). Give the standard deviation exactly and to four decimal places. b) Find the probability that the test-taker answers at least \(10\) questions correctly. Round to four decimal places.

Let \(X\sim\operatorname{Bin}(200,p)\). 1. Find the value of \(p\) that maximizes \(\operatorname{Var}(X)\). Then find the standard deviation for that value of \(p\). 2. Now let \(p=0.15\). Find the mean \(\mu\) and standard deviation \(\sigma\), giving \(\sigma\) exactly and to two decimal places. List all integer values \(k\) in \([\mu-\sigma,\mu+\sigma]\).

A binomial random variable \(X\) has parameters \(n=50\) and \(p=0.4\). 1. Find the mean \(\mu\) and standard deviation \(\sigma\). 2. List all integer values that satisfy \(|X-\mu|\le\sigma\). 3. Find \(P(|X-\mu|\le\sigma)\). Round to four decimal places.

In an electronics manufacturing process, \(5\%\) of components are defective. Assume components are independently classified with this constant defect probability. A random sample of \(400\) components is selected. Let \(X\) be the number of defective components. 1. Find the interval \([\mu-2\sigma,\mu+2\sigma]\). Give the endpoints to two decimal places. 2. Find the probability that \(X\) lies outside this interval. Round to four decimal places. 3. Interpret the result in context, referring to variability around the mean.

A binomial random variable \(X\) has parameters \(n\in\mathbb{N}\), \(n\ge1\), and \(0<p<1\). Its variance is \(60\%\) of its mean. a) Find the success probability \(p\). b) Find \(n\) if the standard deviation is \(\sigma=\sqrt{6}\).

During final inspection at an electronics factory, a sample of \(n\) components is selected. Let \(X\) be the number of defective components, modeled by a binomial distribution. The mean is \(\mu=15\), and the variance is \(\sigma^2=12.75\). a) Find the sample size \(n\) and defect probability \(p\). b) Find the probability that the number of defective components is exactly equal to the mean. Give an exact binomial expression and a decimal approximation to four decimal places.

A spinner has \(k\) equal sections, exactly one of which is a winning section. The spinner is spun \(n\) independent times. Let \(X\) be the number of wins. The binomial random variable has mean \(\mu=5\) and standard deviation \(\sigma=2\). Find \(n\) and \(k\).

A binomial random variable \(X\) has parameters \(n\) and \(p\). a) Find \(n\) and \(p\) if \(E(X)=12\) and \(\operatorname{Var}(X)=9\). b) Use \(\operatorname{Var}(X)=np(1-p)\) to explain why, for a fixed \(p\in(0, 1)\), the variance grows proportionally with \(n\). How does the standard deviation grow as \(n\) increases?

An experiment with success probability \(p=0.2\) is repeated independently \(80\) times. Let \(X\) be the number of successes. a) Find the interval \([\mu-2\sigma,\mu+2\sigma]\) and list the possible integer success counts in the interval. b) Find the probability that the number of successes is within two standard deviations of the mean. Round to four decimal places.

In a large screening program, the probability of a positive test result is \(p=0.04\). Let \(Y\) be the number of positive results among \(500\) independently tested people, and assume \(Y\) is binomial. a) Find the mean \(\mu\) and standard deviation \(\sigma\) of \(Y\). Give \(\sigma\) exactly and to two decimal places. b) Find the smallest and largest possible relative frequencies of positive results, given that \(Y\) is within one standard deviation of its mean. c) For any random variable \(Z\) with mean \(\mu\) and standard deviation \(\sigma\), Chebyshev's inequality states that \(P(\mu-k\sigma<Z<\mu+k\sigma)\ge1-\frac{1}{k^2}.\) Explain what this says for \(k=3\) in the context of this screening program.

In a certain region, \(30\%\) of teenagers play a musical instrument. A school study randomly and independently selects \(50\) teenagers. Would it be considered unusual if \(22\) of them report playing an instrument? Use the two-standard-deviation rule.

A survey reports that \(20\%\) of workers in a large city commute by bicycle. A city planner surveys \(100\) randomly and independently selected workers. Use the two-standard-deviation rule to find the range of bicycle-commuter counts that would be considered consistent with the reported population proportion.

A fair six-sided die is rolled independently \(60\) times, and the number of sixes is recorded. 1. Find the mean \(\mu\) and standard deviation \(\sigma\) of the number of sixes. Give \(\sigma\) exactly and to two decimal places. 2. A result is classified as unusual if it lies outside \([\mu-2\sigma,\mu+2\sigma]\). List all counts of sixes that would raise suspicion that the die is not fair under this rule.

A game has possible payouts of \(\$0\), \(\$2\), and \(\$10\). Let \(X\) be the payout in dollars. It is known that \(P(X=10)=0.05\) and \(P(X=2)=0.25\). a) Find \(P(X=0)\). b) Find the expected payout \(E(X)\). c) Find the standard deviation \(\sigma(X)\).

A prize wheel lands on the Grand Prize section with probability \(p=0.05\). a) How many Grand Prizes are expected in \(800\) spins? b) At an event, the Grand Prize occurred \(35\) times. Suppose this count equals the expected value exactly. How many times was the wheel spun? c) In one trial, the wheel was spun \(2000\) times and landed on Grand Prize \(112\) times. Find the relative frequency and compare it with the theoretical probability.

A four-sided die is rolled \(1000\) times to investigate whether it behaves like a fair die. The observed counts are shown. <table> <thead> <tr> <th>Result \(i\)</th> <th>Observed count</th> </tr> </thead> <tbody> <tr> <td>\(1\)</td> <td>\(210\)</td> </tr> <tr> <td>\(2\)</td> <td>\(290\)</td> </tr> <tr> <td>\(3\)</td> <td>\(240\)</td> </tr> <tr> <td>\(4\)</td> <td>\(260\)</td> </tr> </tbody> </table> 1. Use relative frequencies to estimate the probability distribution of \(X\), the result of one roll. 2. Find the expected value based on the empirical distribution. 3. Find the expected value of an ideal fair four-sided die and compare the two values.

A quality-control study checks \(4000\) newly produced circuit boards for soldering defects. The random variable \(X\) is the number of defects on one board. <table> <thead> <tr> <th>Number of defects \(k\)</th> <th>Observed count</th> </tr> </thead> <tbody> <tr> <td>\(0\)</td> <td>\(3600\)</td> </tr> <tr> <td>\(1\)</td> <td>\(320\)</td> </tr> <tr> <td>\(2\)</td> <td>\(80\)</td> </tr> </tbody> </table> 1. Use relative frequencies to estimate \(P(X=k)\). 2. Estimate the probability that a randomly selected board has at least one defect. 3. Based on the empirical expected number of defects per board, estimate the total number of defects in a shipment of \(500\) boards.

A container holds \(n\) tickets numbered \(1,2,3,\ldots,n\). The sum of all numbers on the tickets is \(1275\). Find the expected number on a randomly selected ticket.

A school carnival offers a game using a spinner with \(20\) equal sections. The game costs \(\$2.50\) to play. Two sections pay \(\$12.00\), five sections pay \(\$4.00\), and the remaining sections pay nothing. Find the expected net gain \(X\) for a player. Decide whether the game is fair, and find the entry fee that would make it fair.

An urn contains \(25\) tickets. One gold ticket pays \(\$20.00\), four silver tickets each pay \(\$5.00\), and the remaining tickets pay nothing. The game costs \(\$3.00\) per draw, and the original ticket distribution is restored after every draw. Find the player’s expected gain or loss per game. What total profit can the operator expect from \(200\) games?

A school carnival game has the following prize schedule. The random variable \(X\) is the payout in dollars. <table> <tr> <th>Payout</th> <td>\(\$0\)</td> <td>\(\$2\)</td> <td>\(\$10\)</td> <td>\(\$50\)</td> </tr> <tr> <th>Probability</th> <td>\(85\%\)</td> <td>\(10\%\)</td> <td>\(4\%\)</td> <td>\(1\%\)</td> </tr> </table> a) Find \(E(X)\). b) The game costs \(\$2.00\) to play. Find the operator’s expected profit per game. c) A new rule requires the expected payout to be at least \(45\%\) of the entry fee. What is the greatest entry fee the operator may charge while keeping the same prize schedule? Give the answer to the nearest cent without exceeding the limit.

An arcade machine pays prizes according to the table. <table> <tr> <th>Payout</th> <td>\(\$0.10\)</td> <td>\(\$0.50\)</td> <td>\(\$2.00\)</td> <td>\(\$5.00\)</td> <td>\(\$H\)</td> </tr> <tr> <th>Probability</th> <td>\(\frac{1}{5}\)</td> <td>\(\frac{1}{20}\)</td> <td>\(\frac{1}{50}\)</td> <td>\(\frac{1}{100}\)</td> <td>\(\frac{1}{500}\)</td> </tr> </table> All other outcomes pay nothing. a) Find the probability that a play has no payout. b) The machine costs \(\$0.50\) per play. Find the jackpot \(H\) so that the expected payout is exactly \(60\%\) of the entry fee.

A spinner has four equal sections, and one section is labeled Win. The spinner is spun independently. a) Find the probability of event \(A\): exactly \(1\) win in \(4\) spins, and event \(B\): exactly \(2\) wins in \(8\) spins. Give exact values and decimal approximations to four decimal places. Are the probabilities equal? b) For \(4\) spins, the expected number of wins is \(1\). Evaluate this claim: “Because \(E(X)=1\), event \(C\), at least one win in \(4\) spins, is certain.” Find \(P(C)\) exactly and to four decimal places as part of your justification.

A random variable \(X\) has a binomial distribution with \(n=5\) and \(p=0.4\). a) Find \(P(X=k)\) for every possible value of \(k\), and display the probability distribution in a table. b) Use only the probabilities in your table to find \(E(X)\) from \(E(X)=\sum_{k=0}^{5}kP(X=k)\).

Consider two independent Bernoulli trials. Each trial is a success with probability \(p\) and a failure with probability \(q=1-p\). Let \(X\) be the number of successes. a) Write the probability distribution of \(X\). b) Find \(E(X)\) in terms of \(p\). c) Use the definition \(\operatorname{Var}(X)=\sum_{i=0}^{2}[x_i-E(X)]^2P(X=x_i)\) to show that \(\operatorname{Var}(X)=2p(1-p)\).

A binomial random variable \(X\) has \(n=250\) and variance \(\operatorname{Var}(X)=40\). a) Find the two possible values of \(p\). b) Find the mean for each value of \(p\). c) Find the greatest possible variance for a binomial random variable with \(n=250\).

A binomial random variable \(X\) has parameters \(n=5\) and \(p=0.5\). Find \(\operatorname{Var}(X)\) using \(\operatorname{Var}(X)=E(X^2)-[E(X)]^2\). Calculate \(E(X)\) and \(E(X^2)\) explicitly from the probability distribution.

A manufacturer performs continuous-use tests on specialized batteries. All batteries fail by \(30\) hours. The table shows the proportion still operating at each time \(t\). <table> <tr> <td>Time \(t\) in hours</td> <td>\(0\)</td> <td>\(5\)</td> <td>\(10\)</td> <td>\(15\)</td> <td>\(20\)</td> <td>\(25\)</td> <td>\(30\)</td> </tr> <tr> <td>Proportion operating</td> <td>\(100\%\)</td> <td>\(95\%\)</td> <td>\(85\%\)</td> <td>\(60\%\)</td> <td>\(30\%\)</td> <td>\(5\%\)</td> <td>\(0\%\)</td> </tr> </table> Estimate the mean battery life by assuming failures are uniformly distributed within each \(5\)-hour interval and using interval midpoints.

A discrete random variable \(X\) can take the values \(1\), \(2\), and \(3\), with \(P(X=1)=a\), \(P(X=2)=0.4\), and \(P(X=3)=0.6-a\), where \(0\le a\le0.6\). a) Find \(a\) when \(E(X)=2.3\). b) Without further calculation, explain whether the expected value increases or decreases as \(a\) increases.

A binomial random variable \(X\) has mean \(\mu=12\) and variance \(\operatorname{Var}(X)=7.2\). a) Find the parameters \(n\) and \(p\). b) Find \(P(X=12)\), the probability that \(X\) is exactly equal to its mean. Round to four decimal places.

A digital prize wheel at a carnival is played about \(12{,}000\) times during a weekend. The prize probabilities and values are shown. <table> <tr> <th>Prize category</th> <th>Probability</th> <th>Prize value</th> </tr> <tr> <td>Grand prize</td> <td>\(\frac{1}{1000}\)</td> <td>\(\$250\)</td> </tr> <tr> <td>Small prize</td> <td>\(\frac{3}{100}\)</td> <td>\(\$10\)</td> </tr> <tr> <td>Consolation prize</td> <td>\(\frac{1}{10}\)</td> <td>\(\$2\)</td> </tr> <tr> <td>No prize</td> <td>\(\frac{869}{1000}\)</td> <td>\(\$0\)</td> </tr> </table> a) Find the expected number of winners in each of the three prize categories. b) Find the expected total value of all prizes awarded during the weekend.

A spinner has \(12\) equal sections labeled \(0\), \(1\), \(2\), and \(5\), as shown. The spinner is spun once, and the random variable \(X\) is the payout in dollars. a) Complete the probability distribution. <table border="1" cellpadding="5" style="border-collapse: collapse; text-align: center; margin: 10px 0;"> <thead> <tr> <th style="background-color: #f2f2f2; padding: 8px;">Payout \(x\)</th> <th style="padding: 8px;">\(\$0\)</th> <th style="padding: 8px;">\(\$1\)</th> <th style="padding: 8px;">\(\$2\)</th> <th style="padding: 8px;">\(\$5\)</th> </tr> </thead> <tbody> <tr> <th style="background-color: #f2f2f2; padding: 8px;">\(P(X=x)\)</th> <td style="padding: 8px;">\(\frac{1}{3}\)</td> <td style="padding: 8px;"></td> <td style="padding: 8px;">\(\frac{1}{4}\)</td> <td style="padding: 8px;"></td> </tr> </tbody> </table> b) Find \(E(X)\). c) The game costs \(\$1.50\) to play. Is the game fair to the player in the long run? Justify your answer.

The urn shown is used in a carnival game. A player pays \(\$1.50\) and draws \(2\) balls without replacement. The payouts are: - Two red balls: \(\$5.00\) - Two blue balls: \(\$2.00\) - The green ball and any other ball: \(\$1.00\) - Any other result: \(\$0.00\) Let \(X\) be the player's net gain, payout minus entry fee. a) Determine the probability distribution of \(X\). b) Calculate \(E(X)\) and decide whether the game is fair.

A carnival game uses the spinner shown. It has \(8\) equal sections, and each number is the payout in dollars. a) Find the probability distribution of the payout \(X\), and calculate \(E(X)\). b) What entry fee makes the game fair to the operator, meaning the operator’s expected profit is \(\$0\) per play? c) What entry fee gives the operator an expected profit of \(\$0.50\) per play?

A carnival game uses the spinner shown, which has \(6\) equal sections. The game costs \(\$3\) to play. A red section labeled \(5\) pays \(\$10\), a blue section labeled \(2\) pays \(\$4\), and a yellow section labeled \(1\) pays nothing. The random variable \(X\) is the player’s net gain, equal to payout minus entry fee. a) Find the probability distribution of \(X\) and display it in a table. b) Find \(E(X)\) and determine whether the game is fair.

At a carnival game, a spinner with \(8\) equal sections is spun once. The amounts shown are the player's net gain or loss in dollars. Let \(X\) be the player's net result from one play. a) Display the probability distribution of \(X\) in a table. b) Find \(E(X)\). Is the game fair? Explain. c) Find \(\sigma(X)\), rounded to the nearest hundredth.

The length of time visitors spend in an art gallery, in hours, is modeled by a continuous random variable \(X\) with density \(f(x)=\frac49x^2-\frac4{27}x^3\quad\text{for }0\le x\le3,\) and \(f(x)=0\) otherwise. a) Show algebraically that \(f\) is a valid probability density function. b) Find \(P(X\le1.5)\). c) Find a formula for the cumulative distribution function \(F(x)\) on \([0,3]\). d) Find the mean time visitors spend in the gallery. Give the result in hours and also in hours and minutes.

Customer wait time \(X\), in minutes, at a service hotline is modeled by \(f(x)=0.2e^{-0.2x}\quad\text{for }x\ge0,\) and \(f(x)=0\) for \(x<0\). The graph of \(f\) is shown on \([0,15]\). a) Describe the graph's intercept, monotonic behavior, and end behavior. b) Find \(P(X\le5)\). c) Find the cumulative distribution function \(F\). d) Find the mean \(\mu=\int_0^\infty x f(x)\,\text{d}x\). Hint: An antiderivative of \(xe^{ax}\) is \(\frac{e^{ax}}{a^2}(ax-1)\).

Let \(X\) represent the fraction of daylight at a location that is obscured by clouds. The density is \(f(x)=6x(1-x)\quad\text{for }0\le x\le1,\) and \(f(x)=0\) otherwise. a) Show that \(f\) is a probability density function. b) Find the mean \(\mu\), and interpret it in context. c) Find \(P(0.25\le X\le0.75)\). d) Find the cumulative distribution function \(F(x)\) on \([0,1]\).

Two bus companies, SprintBus and SteadyLine, serve the same route. Passenger wait time, in minutes, is modeled by density \(f\) for SprintBus and density \(g\) for SteadyLine: \(f(x)=\frac38x^2,\qquad g(x)=\frac38(x-2)^2,\qquad 0\le x\le2.\) Both functions are \(0\) outside \([0,2]\), and their graphs are shown. a) Show algebraically that both functions are valid probability densities. b) For each company, find the probability that a passenger waits longer than \(1\) minute. c) Find the mean wait time for each model. Which company would you prefer when minimizing expected wait time?

A carnival game uses the spinner shown. It has \(10\) equal sections labeled with payout amounts in dollars. The random variable \(X\) is the payout for one play. a) Create a probability distribution table for \(X\). b) Find \(E(X)\). What entry fee makes the game fair? c) The operator wants to charge \(\$5.00\). To keep the game fair, the highest payout, \(\$24.00\), will be replaced by \(\$a\). Find \(a\).

Two tennis players, A and B, play a best-of-three match. The first player to win \(2\) sets wins the match. Player A wins each set independently with probability \(p=0.6\). a) Use the tree diagram to give the probabilities of all possible match sequences. b) Let \(X\) be the number of sets played. Find the probability distribution of \(X\). c) Find \(E(X)\).

A warranty pays \(\$0\) with probability \(0.94\), \(\$200\) with probability \(0.05\), and \(\$1000\) with probability \(0.01\). a) Find the expected payout per warranty. b) Find the variance of the payout. c) Explain why the expected payout is not a prediction of the amount paid on any one warranty.

Four event indicators \(I_1,I_2,I_3,I_4\) have event probabilities \(0.20,0.35,0.50,\) and \(0.80\). Let \(N=I_1+I_2+I_3+I_4\) be the number of events that occur. Find \(E(N)\). Does this calculation require the events to be independent? Explain.

A company knows that \(16\%\) of the parts produced by a certain machine need rework. A quality engineer will inspect a random sample of \(250\) parts and treat the sampled outcomes as independent. Let \(X\) be the number that need rework and let \(\hat p=X/250\). a) Find the mean and standard deviation of \(X\). Give the standard deviation exactly and to four decimal places. b) Find the mean and standard deviation of \(\hat p\). Give the standard deviation exactly and to five decimal places. c) Explain the numerical relationship between the two means and the two standard deviations.

In an automated drilling process, the magnitude of the error \(X\), in millimeters, has density \(f(x)=c(4-x^2)\quad\text{for }0\le x\le2,\) and \(f(x)=0\) otherwise. a) Find the constant \(c\). b) Find the probability that the error is at most \(1.5\,\text{mm}\). c) Find the mean \(E(X)\) and the standard deviation \(\sigma(X)\). Give the standard deviation exactly and to four decimal places.

Let \(X\) be a binomial random variable with parameters \(n\) and \(p\). a) For \(n=50\) and \(p=0.01\), find the numerical values of the variance \(V\) and standard deviation \(\sigma\), and compare them. b) Show that when \(n=2\), the numerical value of the standard deviation is greater than the numerical value of the variance for every \(p\in(0, 1)\).

At a carnival game, a player first rolls a fair four-sided die labeled \(1\) through \(4\). Only a player who rolls a \(1\) may spin a prize wheel with three sections: Grand Prize \((\$60)\), Consolation Prize \((\$15)\), and No Prize \((\$0)\). The No Prize section has a central angle of \(120^\circ\). The Grand Prize and Consolation Prize sections are designed so that the expected prize value per game entry is \(\$5.25\). Find the central angles of the Grand Prize and Consolation Prize sections.

For a binomial random variable \(X\) with parameters \(n\) and \(p\), where \(0<p<1\), let \(\mu\) be the mean and \(\sigma\) the standard deviation. a) Show that \(\sigma^2=\mu(1-p)\). b) Use the result from part a) to show that \(\sigma<\sqrt{\mu}\) for every \(0<p<1\). c) Find \(p\) in terms of \(n\) if the standard deviation is exactly one third of the mean.

For a fixed number of trials \(n\), the variability of a binomial random variable depends on the success probability \(p\). a) Show algebraically that \(V(p)=np(1-p)\) is symmetric about \(p=0.5\) by verifying that \(V(0.5-h)=V(0.5+h)\) for any \(h\in[0, 0.5]\). b) Find the value of \(p\) that maximizes the standard deviation. Express the maximum standard deviation in terms of \(n\).

A binomial random variable \(X\) has parameters \(n\in\mathbb{N}\), \(n\ge1\), and \(0<p<1\). Find all values of \(p\) for which the standard deviation is less than one half of the mean. Express the result in terms of \(n\).

The coefficient of variation of a random variable is the ratio of its standard deviation to its mean. Let \(Z_n\) be the sum of the measurements from \(n\) independent trials, where \(E(Z_n)=8n\) and \(\operatorname{Var}(Z_n)=2n\). Find \(n\) if the coefficient of variation of \(Z_n\) is exactly \(1.25\%\).

Two equally matched basketball teams play a best-of-seven playoff series. The first team to win \(4\) games wins the series, and each team has probability \(0.5\) of winning any game. The random variable \(X\) is the number of games played before the series ends. a) Determine the probability distribution of \(X\). b) Calculate the expected number of games, \(E(X)\).

A nursery studies how long newly planted seedlings survive during their first \(6\) weeks. By the end of week \(6\), none are alive. The table shows the proportion still alive after \(w\) weeks. <table> <tr> <td>Week \(w\)</td> <td>\(0\)</td> <td>\(1\)</td> <td>\(2\)</td> <td>\(3\)</td> <td>\(4\)</td> <td>\(5\)</td> <td>\(6\)</td> </tr> <tr> <td>Proportion alive</td> <td>\(1.00\)</td> <td>\(0.92\)</td> <td>\(0.80\)</td> <td>\(0.55\)</td> <td>\(0.25\)</td> <td>\(0.05\)</td> <td>\(0.00\)</td> </tr> </table> Because the exact time of death within each week is unknown, the mean survival time can only be bounded. Find the lower and upper bounds for the mean survival time in weeks.

The spinner shown has \(6\) equal sections. Before playing, a player chooses to spin exactly \(n\) times. If any spin lands on \(0\), the game ends immediately and the payout is \(\$0\). If none of the \(n\) spins lands on \(0\), the payout is the sum of all numbers spun, in dollars. The expected payout has the form \(E(n)=anb^n\). a) Find \(a\) and \(b\). b) Find \(E(3)\) and \(E(4)\), rounded to the nearest cent. c) Show that two consecutive values of \(n\) give exactly the same expected payout, and identify them.

At a coffee roastery, \(X\) is the amount, in grams, by which a package's fill weight exceeds the required minimum. The continuous random variable \(X\) has density \(f(x)= \begin{cases} kx(4-x)^2,&0\le x\le4,\\ 0,&\text{otherwise}. \end{cases}\) The graph of \(f\) is shown. a) Find \(k\) so that \(f\) is a valid probability density function. b) Find \(P(1\le X\le3)\). c) Estimate the mode from the graph, and then find it exactly by maximizing \(f\) on \([0,4]\). d) The mean is \(1.6\,\text{g}\), and the standard deviation is \(0.8\,\text{g}\). Find the probability that \(X\) is within one standard deviation of the mean.