Calculus an Applied Approach

Ron Larson, David C, Falvo

Chapter 9

Probability and Calculus - all with Video Answers

Educators

Section 1

Discrete Probability

Problem 1

In Exercises $1-4,$ list or describe the elements in the specified set.

$$
\begin{array}{l}{\text { Coin Toss A coin is tossed three times. }} \\ {\text { (a) The sample space } S} \\ {\text { (b) The event } A \text { that at least two heads occur }} \\ {\text { (c) The event } B \text { that no more than one head occurs }}\end{array}
$$

Barsha Rana

Barsha Rana

Numerade Educator

Problem 2

List or describe the elements in the specified set.
$$
\begin{array}{l}{\text { Coin Toss } A \text { coin is tossed. If a head occurs, the coin is }} \\ {\text { tossed again; otherwise, a die is tossed. }} \\ {\text { (a) The sample space } S} \\ {\text { (b) The event } A \text { that } 4,5, \text { or } 6 \text { occurs on the die }} \\ {\text { (c) The event } B \text { that two heads occur }}\end{array}
$$

Barsha Rana

Numerade Educator

Problem 3

List or describe the elements in the specified set.
$$
\begin{array}{l}{\text { Poll Three people are asked their opinions on a political }} \\ {\text { issue. They can answer "In favor" (I), "Opposed" (O), or }} \\ {\text { "Undecided" (U). }} \\ {\text { (a) The sample space } S} \\ {\text { (b) The event } A \text { that at least two people are in favor }} \\ {\text { (c) The event } B \text { that no more than one person is opposed }}\end{array}
$$

Barsha Rana

Numerade Educator

Problem 4

List or describe the elements in the specified set.
$$
\begin{array}{l}{\text { Credit Card Fraud Four cases of credit card fraud }} \\ {\text { are examined. The method of fraud is "stolen card" }} \\ {\text { (S), "counterfeit card" (C), "mail order" (M), or "other"(O). }} \\ {\text { (a) The sample space } S}\end{array}
$$
$$
\begin{array}{l}{\text { (b) The event } A \text { that at least three cases are mail order fraud }} \\ {\text { (c) The event } B \text { that no more than one case is counterfeit }} \\ {\text { card fraud }}\end{array}
$$

Barsha Rana

Numerade Educator

Problem 5

Coin Toss Two coins are tossed. A random variable assigns the number $0,1,$ or 2 to each possible outcome, depending on the number of heads that turn up. Find the frequencies of $0,1,$ and 2 .

Barsha Rana

Numerade Educator

Problem 6

Coin Toss Four coins are tossed. A random variable assigns the number $0,1,2,3,$ or 4 to each possible outcome, depending on the number of heads that turn up. Find the frequencies of $0,1,2,3,$ and $4 .$

Barsha Rana

Numerade Educator

Problem 7

Exam Three students answer a true-false question on an examination. A random variable assigns the number $0,1,2,$ or 3 to each outcome, depending on the number of answers of true among the three students. Find the frequencies of $0,$ $1,2,$ and $3 .$

Barsha Rana

Numerade Educator

Problem 8

Exam Four students answer a true-false question on an examination. A random variable assigns the number $0,1,2,$ $3,$ or 4 to each outcome, depending on the number of answers of true among the four students. Find the frequencies of $0,1,2,3,$ and $4 .$

Barsha Rana

Numerade Educator

Problem 9

Poll Three people have been nominated for president of a college class. From a small poll it is estimated that Jane has a probability of 0.29 of winning and Larry has a probability of $0.47 .$ What is the probability of the third candidate winning the election?

Barsha Rana

Numerade Educator

Problem 10

Random Selection In a class of 72 students, 44 are girls and, of these, 12 are going to college. Of the 28 boys in the class, 9 are going to college. If a student is selected at random from the class, what is the probability that the person chosen is (a) going to college, (b) not going to college, and (c) a girl who is not going to college?

Barsha Rana

Numerade Educator

Problem 11

Quality Control A component of a spacecraft has both a main system and a backup system. The probability of at least one of the systems performing satisfactorily throughout the duration of the flight is $0.9855 .$ What is the probability of both of them failing?

Barsha Rana

Numerade Educator

Problem 12

Random Selection A card is chosen at random from a standard 52-card deck of playing cards. What is the probability that the card will be black and a face card?

Barsha Rana

Numerade Educator

Problem 13

In Exercises 13 and $14,$ find the missing value of the probability distribution.
(Table cant copy)

Barsha Rana

Numerade Educator

Problem 14

Find the missing value of the probability distribution.
(Table cant copy)

Barsha Rana

Numerade Educator

Problem 15

In Exercises $15-18,$ determine whether the table represents a probability distribution. If it is a probability distribution, sketch its graph. If it is not a probability distribution, state any properties that are not satisfied.
(Table cant copy)

Barsha Rana

Numerade Educator

Problem 16

Determine whether the table represents a probability distribution. If it is a probability distribution, sketch its graph. If it is not a probability distribution, state any properties that are not satisfied.
(Table cant copy)

Barsha Rana

Numerade Educator

Problem 17

Determine whether the table represents a probability distribution. If it is a probability distribution, sketch its graph. If it is not a probability distribution, state any properties that are not satisfied.
(Table cant copy)

Barsha Rana

Numerade Educator

Problem 18

Determine whether the table represents a probability distribution. If it is a probability distribution, sketch its graph. If it is not a probability distribution, state any properties that are not satisfied.
(Table cant copy)

Barsha Rana

Numerade Educator

Problem 19

In Exercises $19-22,$ sketch a graph of the probability distribution and find the required probabilities.
$$
\begin{array}{l}{\text { (a) } P(1 \leq x \leq 3)} \\ {\text { (b) } P(x \geq 2)}\end{array}
$$

Barsha Rana

Numerade Educator

Problem 20

Sketch a graph of the probability distribution and find the required probabilities.
$$
\begin{array}{l}{\text { (a) } P(x \leq 2)} \\ {\text { (b) } P(x>2)}\end{array}
$$

Barsha Rana

Numerade Educator

Problem 21

Sketch a graph of the probability distribution and find the required probabilities.
$$
\begin{array}{l}{\text { (a) } P(x \leq 3)} \\ {\text { (b) } P(x>3)}\end{array}
$$

Barsha Rana

Numerade Educator

Problem 22

Sketch a graph of the probability distribution and find the required probabilities.
$$
\begin{array}{l}{\text { (a) } P(1 \leq x \leq 2)} \\ {\text { (b) } P(x<2)}\end{array}
$$

Barsha Rana

Numerade Educator

Problem 23

Biology Consider a couple who have four children. Assume that it is equally likely that each child is a girl or a boy.
$$
\begin{array}{l}{\text { (a) Complete the set to form the sample space consisting }} \\ {\text { of } 16 \text { elements. }} \\ {S=\{g g g g, g g g b, g g b g, \ldots\}} \\ {\text { (b) Complete the table, in which the random variable } x \text { is }} \\ {\text { the number of girls in the family. }}\end{array}
$$
$$
\begin{array}{l}{\text { (c) Use the table in part (b) to sketch the graph of the }} \\ {\text { probability distribution. }} \\ {\text { (d) Use the table in part (b) to find the probability that at }} \\ {\text { least one of the children is a boy. }}\end{array}
$$

Barsha Rana

Numerade Educator

Problem 24

Die Toss Consider the experiment of tossing a 12 -sided die twice.
$$
\begin{array}{l}{\text { (a) Complete the set to form the sample space of } 144} \\ {\text { elements. Note that each element is an ordered pair in }} \\ {\text { which the entries are the numbers of points on the first }} \\ {\text { and second tosses, respectively. }} \\ {S=\{(1,1),(1,2), \ldots,(2,1),(2,2), \ldots\}} \\ {\text { (b) Complete the table, in which the random variable } x \text { is }} \\ {\text { the sum of the number of points. }}\end{array}
$$
$$
\begin{array}{l}{\text { (c) Use the table in part (b) to sketch the graph of the }} \\ {\text { probability distribution. }} \\ {\text { (d) Use the table in part (b) to find } P(15 \leq x \leq 19) \text { . }}\end{array}
$$

Lucas Finney

Numerade Educator

Problem 25

In Exercises $25-28,$ find $E(x), V(x),$ and $\sigma$ for the given probability distribution.
(Table cant copy)

Barsha Rana

Numerade Educator

Problem 26

Find $E(x), V(x),$ and $\sigma$ for the given probability distribution.
(Table cant copy)

Barsha Rana

Numerade Educator

Problem 27

Find $E(x), V(x),$ and $\sigma$ for the given probability distribution.
(Table cant copy)

Barsha Rana

Numerade Educator

Problem 28

Find $E(x), V(x),$ and $\sigma$ for the given probability distribution.
(Table cant copy)

Barsha Rana

Numerade Educator

Problem 29

In Exercises 29 and $30,$ find the mean and variance of the discrete random variable $x .$
$$
\begin{array}{l}{\text { Die Toss } x \text { is (a) the number of points when a four-sided }} \\ {\text { die is tossed once and (b) the sum of the points when the }} \\ {\text { four-sided die is tossed twice. }}\end{array}
$$

Barsha Rana

Numerade Educator

Problem 30

Find the mean and variance of the discrete random variable $x .$
$$
\begin{array}{l}{\text { Coin Toss } x \text { is the number of heads when a coin is tossed }} \\ {\text { four times. }}\end{array}
$$

Barsha Rana

Numerade Educator

Problem 31

Revenue A publishing company introduces a new weekly magazine that sells for $\$ 4.95 dollar on the newsstand. The marketing group of the company estimates that sales $x$ (in thousands) will be approximated by the following probability function.
$$
\begin{array}{l}{\text { (a) Find } E(x) \text { and } \sigma .} \\ {\text { (b) Find the expected revenue. }}\end{array}
$$

Barsha Rana

Numerade Educator

Problem 32

Personal Income The probability distribution of the random variable $x,$ the annual income of a family (in thousands of dollars) in a certain section of a large city, is shown in the table.
Find $E(x)$ and $\sigma$
(Table cant copy)

Lucas Finney

Numerade Educator

Problem 33

Insurance An insurance company needs to determine the annual premium required to break even on fire protection policies with a face value of 90,000 dollar If $x$ is the claim size on these policies and the analysis is restricted to the losses 30,000, 60,000, dollar and 90,000,dollar then the probability distribution of $x$ is as shown in the table. What premium should customers be charged for the company to break even?
(Table cant copy)

Lucas Finney

Numerade Educator

Problem 34

Insurance An insurance company needs to determine the annual premium required to break even for collision protection for cars with a value of 10,000 .dollar If $x$ is the claim size on these policies and the analysis is restricted to the losses 1000, 5000, dollar and 10,000, dollar then the probability distribution of $x$ is as shown in the table. What premium should customers be charged for the company to break even?

Lucas Finney

Numerade Educator

Problem 35

Games of Chance If $x$ is the net gain to a player in a game of chance, then $E(x)$ is usually negative. This value gives the average amount per game the player can expect to lose over the long run. In Exercises 35 and $36,$ find the expected net gain to the player for one play of the specified game.
$$
\begin{array}{l}{\text { In roulette, the wheel has the } 38 \text { numbers } 00,0,1,2, \ldots} \\ {34,35, \text { and } 36, \text { marked on equally spaced slots. If a player }} \\ {\text { bets } \$ 1 \text { on a number and wins, then the player keeps the }} \\ {\text { dollar and receives an additional } \$ 35 . \text { Otherwise, the dollar }} \\ {\text { is lost. }}\end{array}
$$

Lucas Finney

Numerade Educator

Problem 36

Games of Chance If $x$ is the net gain to a player in a game of chance, then $E(x)$ is usually negative. This value gives the average amount per game the player can expect to lose over the long run. In Exercises 35 and $36,$ find the expected net gain to the player for one play of the specified game.
$$
\begin{array}{l}{\text { A service organization is selling } \$ 2 \text { raffle tickets as part of }} \\ {\text { a fundraising program. The first prize is a boat valued at }} \\ {\$ 2950, \text { and the second prize is a camping tent valued at }} \\ {\$ 400 . \text { In addition to the first and second prizes, there as }} \\ {\$ 20 \text { gift certificates to be awarded. The number of tickets }} \\ {\text { sold is } 3000 \text { . }}\end{array}
$$

Lucas Finney

Numerade Educator

Problem 37

Market Analysis After considerable market study, a sporting goods company has decided on two possible cities in which to open a new store. Management estimates that city 1 will yield 20 dollar million in revenues if successful and will lose 4 million dollar if not, whereas city 2 will yield 50 million dollar in revenues if successful and lose 9 million dollar if not. City 1 has a 0.3 probability of being successful and city 2 has a 0.2 probability of being successful. In which city should the sporting goods company open the new store with respect to the expected return from each store?

Lucas Finney

Numerade Educator

Problem 38

Repeat Exercise 37 for the case in which the probabilities of city 1 and city 2 being successful are 0.4 and $0.25,$ respectively.

Lucas Finney

Numerade Educator

Problem 39

Health The table shows the probability distribution of the numbers of AIDS cases diagnosed in the United States in 2005 by age group.
$$
\begin{array}{l}{\text { (a) Sketch the probability distribution. }} \\ {\text { (b) Find the probability that an individual diagnosed with }} \\ {\text { AIDS was from } 15 \text { to } 44 \text { years of age. }} \\ {\text { (c) Find the probability that an individual diagnosed with }} \\ {\text { AIDS was at least } 35 \text { years of age. }} \\ {\text { (d) Find the probability that an individual diagnosed with }} \\ {\text { AIDS was at most } 24 \text { years of age. }}\end{array}
$$

Lucas Finney

Numerade Educator

Problem 40

Education The table gives the probability distribution of the educational attainments of people in the United States in $2005,$ ages 25 years old and over, where $x=0$ represents no high school diploma, $x=1$ represents a high school diploma, $x=2$ represents some college, $x=3$ represents an associate's degree, $x=4$ represents a bachelor's degree, and $x=5$ represents an advanced degree.
$$
\begin{array}{l}{\text { (a) Sketch the probability distribution. }} \\ {\text { (b) Determine } E(x), V(x), \text { and } \sigma \text { . Explain the meanings of }} \\ {\text { these values. }}\end{array}
$$

Lucas Finney

Numerade Educator

Problem 41

Athletics A baseball fan examined the record of a favorite baseball player's performance during his last 50 games. The numbers of games in which the player had zero, one, two, three, and four hits are recorded in the table shown below.
$$
\begin{array}{l}{\text { (a) Complete the table below, where } x \text { is the number of }} \\ {\text { hits. }}\end{array}
$$
$$
\begin{array}{l}{\text { (b) Use the table in part (a) to sketch the graph of the }} \\ {\text { probability distribution. }} \\ {\text { (c) Use the table in part (a) to find } P(1 \leq x \leq 3)} \\ {\text { (d) Determine } E(x), V(x), \text { and } \sigma \text { . Explain your results. }}\end{array}
$$

Lucas Finney

Numerade Educator

Problem 42

Economics: Investment Suppose you are trying to make a decision about how to invest 10,000 dollar over the next year. One option is a low-risk bank deposit paying 5 dollar interest per year. The other is a high-risk corporate stock with a 5 dollar dividend, plus a 50 dollar % chance of a 30 dollar % price decline and a 50 dollar chance of a 30 dollar price increase. Determine the expected value of each option and choose one of the options. Explain your choice. How would your decision change if the corporate stock offered a 20 dollar dividend instead of a 5 dollar dividend?

Bryan Meares

Numerade Educator

Problem 43

Extended Application To work an extended application analyzing the health insurance coverage status of people in the United States by age, visit this text's website at college. hmco.com.

Carson Merrill

Numerade Educator