Chapter 8

The probability distribution of a random variable \(X\) is given. Compute the mean, variance, and standard deviation of \(X\). $$\begin{array}{lllll}\hline x & 1 & 2 & 3 & 4 \\\\\hline P(X=x) & .4 & .3 & .2 & .1 \\\\\hline\end{array}$$

During the first year at a university that uses a 4 -point grading system, a freshman took ten 3 -credit courses and received two As, three Bs, four Cs, and one \(D\). a. Compute this student's grade-point average. b. Let the random variable \(X\) denote the number of points corresponding to a given letter grade. Find the probability distribution of the random variable \(X\) and compute \(E(X)\), the expected value of \(X\).

Three balls are selected at random without replacement from an urn containing four green balls and six red balls. Let the random variable \(X\) denote the number of green balls drawn. a. List the outcomes of the experiment. b. Find the value assigned to each outcome of the experiment by the random variable \(X\). c. Find the event consisting of the outcomes to which a value of 3 has been assigned by \(X\).

Let \(A\) and \(B\) be two events in a sample space \(S\) such that \(P(A)=.6, P(B)=.5\), and \(P(A \cap B)=.2\). Find a. \(P(A \mid B)\) b. \(P(B \mid A)\)

Find the probability of the given event. The coin lands heads all five times.

The probability distribution of a random variable \(X\) is given. Compute the mean, variance, and standard deviation of \(X\). $$\begin{array}{lccccc}\hline x & -4 & -2 & 0 & 2 & 4 \\\\\hline P(X=x) & .1 & .2 & .3 & .1 & .3 \\\\\hline\end{array}$$

Records kept by the chief dietitian at the university cafeteria over a 30-wk period show the following weekly consumption of milk (in gallons). $$\begin{array}{lccccc}\hline \text { Milk } & 200 & 205 & 210 & 215 & 220 \\\\\hline \text { Weeks } & 3 & 4 & 6 & 5 & 4 \\\\\hline\end{array}$$ $$\begin{array}{lcccc}\hline \text { Milk } & 225 & 230 & 235 & 240 \\\\\hline \text { Weeks } & 3 & 2 & 2 & 1 \\ \hline\end{array}$$ a. Find the average number of gallons of milk consumed per week in the cafeteria. b. Let the random variable \(X\) denote the number of gallons of milk consumed in a week at the cafeteria. Find the probability distribution of the random variable \(X\) and compute \(E(X)\), the expected value of \(X\).

A coin is tossed four times. Let the random variable \(X\) denote the number of tails that occur. a. List the outcomes of the experiment. b. Find the value assigned to each outcome of the experiment by the random variable \(X\). c. Find the event consisting of the outcomes to which a value of 2 has been assigned by \(X\).

Find the probability of the given event. The coin lands heads exactly once.

The probability distribution of a random variable \(X\) is given. Compute the mean, variance, and standard deviation of \(X\). $$\begin{array}{lccccc}\hline \boldsymbol{x} & -2 & -1 & 0 & 1 & 2 \\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & 1 / 16 & 4 / 16 & 6 / 16 & 4 / 16 & 1 / 16 \\\\\hline\end{array}$$

Find the expected value of a random variable \(X\) having the following probability distribution: $$\begin{array}{lcccccc}\hline \boldsymbol{x} & -5 & -1 & 0 & 1 & 5 & 8 \\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & .12 & .16 & .28 & .22 & .12 & .10 \\\\\hline\end{array}$$

A die is rolled repeatedly until a 6 falls uppermost. Let the random variable \(X\) denote the number of times the die is rolled. What are the values that \(X\) may assume?

Let \(A\) and \(B\) be two events in a sample space \(S\) such that \(P(A)=.6\) and \(P(B \mid A)=.5\). Find \(P(A \cap B)\).

The probability distribution of a random variable \(X\) is given. Compute the mean, variance, and standard deviation of \(X\). $$\begin{array}{lllllll}\hline \boldsymbol{x} & 10 & 11 & 12 & 13 & 14 & 15 \\\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & 1 / 8 & 2 / 8 & 1 / 8 & 2 / 8 & 1 / 8 & 1 / 8 \\\\\hline\end{array}$$

Find the expected value of a random variable \(X\) having the following probability distribution: $$\begin{array}{lllllll}\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline P(X=x) & \frac{1}{8} & \frac{1}{4} & \frac{3}{16} & \frac{1}{4} & \frac{1}{16} & \frac{1}{8} \\\\\hline\end{array}$$

Cards are selected one at a time without replacement from a well-shuffled deck of 52 cards until an ace is drawn. Let \(X\) denote the random variable that gives the number of cards drawn. What values may \(X\) assume?

The probability distribution of a random variable \(X\) is given. Compute the mean, variance, and standard deviation of \(X\). $$\begin{array}{lccccc}\hline \boldsymbol{x} & 430 & 480 & 520 & 565 & 580 \\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & .1 & .2 & .4 & .2 & .1 \end{array}$$

The daily earnings \(X\) of an employee who works on a commission basis are given by the following probability distribution. Find the employee's expected earnings. $$\begin{array}{lllll}\hline \boldsymbol{x}(\boldsymbol{i n} \$) & 0 & 25 & 50 & 75 \\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & .07 & .12 & .17 & .14 \\\ \hline\end{array}$$ $$\begin{array}{lccc}\hline x(\text { in } \$) & 100 & 125 & 150 \\ \hline P(X=x) & .28 & .18 & .04 \\\\\hline \end{array}$$

Let \(X\) denote the random variable that gives the sum of the faces that fall uppermost when two fair dice are rolled. Find \(P(X=7)\).

Determine whether the events \(A\) and \(B\) are independent. \(P(A)=.3, P(B)=.6, P(A \cap B)=.18\)

The probability distribution of a random variable \(X\) is given. Compute the mean, variance, and standard deviation of \(X\). $$\begin{array}{lccccc}\hline x & -198 & -195 & -193 & -188 & -185 \\ \hline P(X=x) & .15 & .30 & .10 & .25 & .20 \\ \hline\end{array}$$

In a four-child family, what is the expected number of boys? (Assume that the probability of a boy being born is the same as the probability of a girl being born.)

Two cards are drawn from a well-shuffled deck of 52 playing cards. Let \(X\) denote the number of aces drawn. Find \(P(X=2)\)

Determine whether the events \(A\) and \(B\) are independent. \(P(A)=.6, P(B)=.8, P(A \cap B)=.2\)

Based on past experience, the manager of the VideoRama Store has compiled the following table, which gives the probabilities that a customer who enters the VideoRama Store will buy \(0,1,2,3\), or 4 DVDs. How many DVDs can a customer entering this store be expected to buy? $$\begin{array}{lccccc} \hline \text { DVDs } & 0 & 1 & 2 & 3 & 4 \\\\\hline \text { Probability } & .42 & .36 & .14 & .05 & .03 \\ \hline\end{array}$$

Give the range of values that the random variable \(X\) may assume and classify the random variable as finite discrete, infinite discrete, or continuous. \(X=\) The number of times a die is thrown until a 2 appears

If a sample of three batteries is selected from a lot of ten, of which two are defective, what is the expected number of defective batteries?

Give the range of values that the random variable \(X\) may assume and classify the random variable as finite discrete, infinite discrete, or continuous. \(X=\) The number of defective watches in a sample of eight watches

The number of accidents that occur at a certain intersection known as "Five Corners" on a Friday afternoon between the hours of 3 p.m. and 6 p.m., along with the corresponding probabilities, are shown in the following table. Find the expected number of accidents during the period in question. $$\begin{array}{lccccc}\hline \text { Accidents } & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Probability } & .935 & .030 & .020 & .010 & .005 \\\\\hline\end{array}$$

Give the range of values that the random variable \(X\) may assume and classify the random variable as finite discrete, infinite discrete, or continuous. \(X=\) The distance in miles a commuter travels to work

If \(A\) and \(B\) are independent events, \(P(A)=.4\), and \(P(B)=.6\), find a. \(P(A \cap B)\) b. \(P(A \cup B)\)

The owner of a newsstand in a college community estimates the weekly demand for a certain magazine as follows: $$\begin{array}{lcccccc}\hline \begin{array}{l}\text { Quantity } \\\\\text { Demanded }\end{array} & 10 & 11 & 12 & 13 & 14 & 15 \\\\\hline \text { Probability } & .05 & .15 & .25 & .30 & .20 & .05 \\ \hline\end{array}$$ Find the number of issues of the magazine that the newsstand owner can expect to sell per week.

Give the range of values that the random variable \(X\) may assume and classify the random variable as finite discrete, infinite discrete, or continuous. \(X=\) The number of hours a child watches television on a given day

If \(A\) and \(B\) are independent events, \(P(A)=.35\), and \(P(B)=.45\), find a. \(P(A \cap B)\) b. \(P(A \cup B)\)

An experiment consists of rolling an eight-sided die (numbered 1 through 8 ) and observing the number that appears uppermost. Find the mean and variance of this experiment.

A bank has two automatic tellers at its main office and two at each of its three branches. The number of machines that break down on a given day, along with the corresponding probabilities, are shown in the following table. $$\begin{array}{l}\text { Machines That } \\\\\text { Break Down } & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Probability } & .43 & .19 & .12 & .09 & .04 \\ \hline\end{array}$$ $$\begin{array}{lcccc}\hline \text { Machines That } & & & & \\ \text { Break Down } & 5 & 6 & 7 & 8 \\\\\hline \text { Probability } & .03 & .03 & .02 & .05 \\\\\hline\end{array}$$ Find the expected number of machines that will break down on a given day.

Give the range of values that the random variable \(X\) may assume and classify the random variable as finite discrete, infinite discrete, or continuous. \(X=\) The number of times an accountant takes the CPA examination before passing

The minimum age requirement for a regular driver's license differs from state to state. The frequency distribution for this age requirement in the 50 states is given in the following table: $$\begin{array}{lllllll}\hline \text { Minimum } & & & & & & \\\\\text { Age } & 15 & 16 & 17 & 18 & 19 & 21 \\ \hline \text { Frequency of } & & & & & & \\\\\text { Occurrence } & 1 & 15 & 4 & 28 & 1 & 1 \\ \hline\end{array}$$ a. Describe a random variable \(X\) that is associated with these data. b. Find the probability distribution for the random variable \(X\). c. Compute the mean, variance, and standard deviation of \(X\).

The management of the Cambridge Company has projected the sales of its products (in millions of dollars) for the upcoming year, with the associated probabilities shown in the following table: $$\begin{array}{lcccccc}\hline \text { Sales } & 20 & 22 & 24 & 26 & 28 & 30 \\\\\hline \text { Probability } & .05 & .10 & .35 & .30 & .15 & .05 \\\\\hline\end{array}$$ What does the management expect the sales to be next year?

Give the range of values that the random variable \(X\) may assume and classify the random variable as finite discrete, infinite discrete, or continuous. \(X=\) The number of boys in a four-child family

Refer to the following experiment: Two cards are drawn in succession without replacement from a standard deck of 52 cards. What is the probability that the first card is a heart given that the second card is a diamond?

The birthrates in the United States for the years \(1991-2000\) are given in the following table. (The birthrate is the number of live births/1000 population.) $$\begin{array}{lllll}\hline \text { Year } & 1991 & 1992 & 1993 & 1994 \\\\\hline \text { Birthrate } & 16.3 & 15.9 & 15.5 & 15.2 \\\\\hline\end{array}$$ $$\begin{array}{llll}\hline \text { Year } & 1995 & 1996 & 1997 \\ \hline \text { Birthrate } & 14.8 & 14.7 & 14.5 \\ \hline\end{array}$$ $$\begin{array}{llll}\hline \text { Year } & 1998 & 1999 & 2000 \\\\\hline \text { Birthrate } & 14.6 & 14.5 & 14.7 \\ \hline\end{array}$$ a. Describe a random variable \(X\) that is associated with these data. b. Find the probability distribution for the random variable \(X\). c. Compute the mean, variance, and standard deviation of \(X\).

A panel of 50 economists was asked to predict the average prime interest rate for the upcoming year. The results of the survey follow: $$\begin{array}{lcccccc}\hline \text { Interest Rate, } \% & 4.9 & 5.0 & 5.1 & 5.2 & 5.3 & 5.4 \\\\\hline \text { Economists } & 3 & 8 & 12 & 14 &8 & 5 \\ \hline\end{array}$$ Based on this survey, what does the panel expect the average prime interest rate to be next year?

The probability distribution of the random variable \(X\) is shown in the accompanying table: $$\begin{array}{lccccccc}\hline \boldsymbol{x} & -10 & -5 & 0 & 5 & 10 & 15 & 20 \\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & .20 & .15 & .05 & .1 & .25 & .1 & .15 \\ \hline\end{array}$$ Find a. \(P(X=-10)\) b. \(P(X \geq 5)\) c. \(P(-5 \leq X \leq 5)\) d. \(P(X \leq 20)\)

Refer to the following experiment: Two cards are drawn in succession without replacement from a standard deck of 52 cards. What is the probability that the first card is a jack given that the second card is an ace?

Find the probability that a family with three children will have the given composition. Two boys and one girl

Paul Hunt is considering two business ventures. The anticipated returns (in thousands of dollars) of each venture are described by the following probability distributions: $$\begin{array}{l}\text { Venture } \mathrm{A}\\\\\begin{array}{cc} \hline \text { Earnings } & \text { Probability } \\\\\hline-20 & .3 \\ \hline 40 & .4 \\\\\hline 50 & .3 \\ \hline\end{array}\end{array}$$ $$\begin{array}{l}\text { Venture } \overline{\mathrm{B}}\\\\\begin{array}{cc} \hline \text { Earnings } & \text { Probability } \\\\\hline-15 & .2 \\ \hline 30 & .5 \\\\\hline 40 & 3 \\ \hline\end{array}\end{array}$$ a. Compute the mean and variance for each venture. b. Which investment would provide Paul with the higher expected return (the greater mean)? c. In which investment would the element of risk be less (that is, which probability distribution has the smaller variance)?

A panel of 64 economists was asked to predict the average unemployment rate for the upcoming year. The results of the survey follow: $$\begin{array}{lccccccc} \hline \text { Unemployment } & & & & & & & \\ \text { Rate, \% } & 4.5 & 4.6 & 4.7 & 4.8 & 4.9 & 5.0 & 5.1 \\ \hline \text { Economists } & 2 & 4 & 8 & 20 & 14 & 12 & 4 \\ \hline\end{array}$$ Based on this survey, what does the panel expect the average unemployment rate to be next year?

The probability distribution of the random variable \(X\) is shown in the accompanying table: $$\begin{array}{lcccccc} \hline x & -5 & -3 & -2 & 0 & 2 & 3 \\ \hline P(X=x) & .17 & .13 & .33 & .16 & 11 & .10 \\ \hline\end{array}$$ Find a. \(P(X \leq 0)\) b. \(P(X \leq-3)\) c. \(P(-2 \leq X \leq 2)\)

Refer to the following experiment: Two cards are drawn in succession without replacement from a standard deck of 52 cards. What is the probability that the first card is a face card given that the second card is an ace?