Deck 8: Probability Distributions and Statistics

Question 1

A coin is tossed four times. Let the random variable X denote the number of heads that occur. Let H denotes the outcome "heads" and T denotes the outcome "tails". Find the value assigned to each outcome of the experiment by the random variable X. ​

A)
B)
C)
D)

Answer 1

Question 2

An examination consisting of ten true-or-false questions was taken by a class of 100 students. The probability distribution of the random variable X, where X denotes the number of questions answered correctly by a randomly chosen student, is represented by the accompanying histogram. The rectangle with base centered on the number 8 is missing. What should be the height of this rectangle? ​ ​

A) 0.35
B) 0.25
C) 0.4
D) 0.2

Answer 2

0.35

Question 3

A die is cast repeatedly until a 4 falls uppermost. Let the random variable X denote the number of times the die is cast. What are the values that X may assume? ​

A) X may assume the values in the set S = {1, 2, 3, ...}
B) X may assume the values in the set S = {4}
C) X may assume the values in the set S = {1, 2, 3, 4, 5, 6}

Answer 3

X may assume the values in the set S = {1, 2, 3, ...}

Question 4

An examination consisting of ten true-or-false questions was taken by a class of 100 students. The probability distribution of the random variable X, where X denotes the number of questions answered correctly by a randomly chosen student, is represented by the accompanying histogram. The rectangle with base centered on the number 8 is missing. What should be the height of this rectangle? ​ ​

A) 0.15
B) 0.3
C) 0.25
D) 0.1

Question 5

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 a commuter travels to work
​

A) . The random variable is infinite discrete.
B) . The random variable is continuous.
C) . The random variable is finite discrete.

Question 6

Three balls are selected at random without replacement from an urn containing four white balls and six red balls. Let the random variable X denote the number of white balls drawn. List the outcomes of the experiment.

A) WWW, WWR, WRW, RWW, WRR, RWR, RRW
B) WWW, WWR, WRW, RWW
C) WWW, WWR, WRW, RWW, WRR, RWR, RRW, RRR
D) WRR, RWR, RRW, RRR

Question 7

The probability distribution of the random variable X is shown in the accompanying table: Find , and .

A)
B)
C)
D)

Question 8

The probability distribution of the random variable X is shown in the accompanying table.

A) 0.35
B) 0.6
C) 0.55
D) 0.45
E) 0.3

Question 9

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? ​

A) X may assume the values in the set S = {1, 2, ...,52}
B) X may assume the values in the set S = {1, 2, ...,47}
C) X may assume the values in the set S = {1, 2, ...,48}
D) X may assume the values in the set S = {1, 2, ...,49}

Question 10

Let X denote the random variable that gives the sum of the faces that fall uppermost when two fair dice are cast. Find P ( X = 5 ). ​

A)
B)
C)
D)

Question 11

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The area of a histogram associated with a probability distribution is a number between 0 and 1.

A) False. The area is exactly equal to one.
B) True. This follows from the definition.

Question 12

Three cards are drawn from a well-shuffled deck of 52 playing cards. Let X denote the number of aces drawn. Find . ​

A)
B)
C)
D)
E)

Question 13

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
​

A) , Finite Continuous
B) Any positive integer, Infinite discrete
C) , discrete
D) , Finite discrete
E) , Continuous

Question 14

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

A) Any integer; infinite discrete
B) 0 ≤ x < ∞; continuous
C) Any positive integer; infinite discrete
D) 0 ≤ x < 24; continuous
E) Any positive integer; infinite continuous

Question 15

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 three-child family
​

A) {1, 2, 3}. The variable is finite discrete.
B) {1, 2, 3}. The variable is continuous.
C) {1, 2, 3}. The variable is infinite discrete.

Question 16

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 3 appears.
​

A) X may assume the values of any positive integer. The random variable is finite discrete.
B) X may assume the values of any positive integer. The random variable is continuous.
C) X may assume the values in the set S = {1, 2, 3, 4, 5, 6}. The random variable is finite discrete.
D) X may assume the values of any positive integer. The random variable is infinite discrete.

Question 17

An examination consisting of ten true-or-false questions was taken by a class of 100 students. The probability distribution of the random variable X, where X denotes the number of questions answered correctly by a randomly chosen student, is represented by the accompanying histogram. The rectangle with base centered on the number 8 is missing. What should be the height of this rectangle? ​ ​

A) 0.3
B) 0.35
C) 0.45
D) 0.5

Question 18

Find the condition on the numbers a and b such that the table below gives the probability distribution of the random variable X. ​

A)
B)
C)
D)
E)

Question 19

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 four watches.
​

A) {0,1,2,3,4}; The random variable is finite discrete
B) {0,1,2,3,4}; The random variable is infinite discrete
C) X may assume the values of any positive integer. The random variable is infinite discrete.
D) X may assume the values of any positive integer. The random variable is continuous.

Question 20

Determine whether the table gives the probability distribution of the random variable X. Explain your answer.

A) No, the sum of the probability assigned to the value of the random variable X is greater than 1.
B) No, the sum of the probability assigned to the value of the random variable X is not equal to 1.
C) No, the probability assigned to a value of the random variable X cannot be negative.
D) Yes, the sum of the probability assigned to the value of the random variable X is equal to 1.
E) No, the sum of the probability assigned to the value of the random variable X is less than 1.

Question 21

Suppose a probability distribution of a random variable X is represented by the accompanying histogram. Shade that part of the histogram whose area gives the probability .

A)
B)
C)
D)
E)

Question 22

An examination consisting of ten true-or-false questions was taken by a class of 100 students. The probability distribution of the random variable X, where X denotes the number of questions answered correctly by a randomly chosen student, is represented by the accompanying histogram. The rectangle with base centered on the number 8 is missing. What should be the height of this rectangle?
​ ​
height = __________

Question 23

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. List the outcomes of the experiment.

Question 24

The probability distribution of the random variable X is shown in the accompanying table.
Find the following, rounded to two decimal places, if necessary. __________ __________ __________ __________

Question 25

After the private screening of a new television pilot, audience members were asked to rate the new show on a scale of 1 to 10 (10 being the highest rating). From a group of 140 people, the accompanying responses were obtained. ​ ​
Let the random variable X denote the rating given to the show by a randomly chosen audience member. Find the probability distribution associated with these data. Round answer to the nearest thousandth.

A) ​
B) ​ ​
C) ​ ​
D) ​ ​
E) ​ ​

Question 26

The probability distribution of the random variable X is shown in the accompanying table. Find the following, rounded to two decimal places if necessary. __________ __________ __________

Question 27

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 a commuter travels to work

Question 28

Suppose a probability distribution of a random variable X is represented by the accompanying histogram. Shade that part of the histogram whose area gives the probability . ​ ​

A) ​
B)
C)
D)
E)

Question 29

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

Question 30

Four cards are drawn from a well-shuffled deck of 52 playing cards. Let X denote the number of aces drawn. Find .

Question 31

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 5 appears.

Question 32

Suppose a probability distribution of a random variable X is represented by the accompanying histogram. Shade that part of the histogram whose area gives the probability . ​ ​

A) ​
B)
C)
D)
E)

Question 33

A survey was conducted by the Public Housing Authority in a certain community among 1,000 families to determine the distribution of families by size. The results follow. Find the probability distribution of the random variable X, where X denotes the number of persons in a randomly chosen family. ​

A) ​ ​
B) ​ ​
C) ​
D) ​
E) ​

Question 34

A die is cast repeatedly until a 4 falls uppermost. Let the random variable X denote the number of times the die is cast. What are the values that X may assume?

Question 35

The rates paid by 30 financial institutions on a certain day for money market deposit accounts are shown in the accompanying table. Let the random variable X denote the interest paid by a randomly chosen financial institution on its money market deposit accounts and find the probability distribution associated with these data.

Question 36

An examination consisting of ten true-or-false questions was taken by a class of 100 students. The probability distribution of the random variable X, where X denotes the number of questions answered correctly by a randomly chosen student, is represented by the accompanying histogram. The rectangle with base centered on the number 8 is missing. What should be the height of this rectangle?
​ ​
height = __________

Question 37

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 six watches.

Question 38

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

Question 39

The following table gives the 2002 age distribution of U.S. population: Let the random variable X denote a randomly chosen age group within the population. What percentage of the U.S. population is between 20 to 44 years old, inclusive?

A) 6.06%
B) 24.07%
C) 63.82%
D) 30.12%
E) 36.18%

Question 40

An examination consisting of ten true-or-false questions was taken by a class of 100 students. The probability distribution of the random variable X, where X denotes the number of questions answered correctly by a randomly chosen student, is represented by the accompanying histogram. The rectangle with base centered on the number 8 is missing. What should be the height of this rectangle?
​ ​
height = __________

Question 41

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 videocassettes. How many videocassettes can a customer entering this store be expected to buy?

A) E = 0.56
B) E = 0.66
C) E = 0.91
D) E = 0.86

Question 42

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: Find the expected number of machines that will break down on a given day.

A) E = 1.76
B) E = 1.54
C) E = 0.71
D) E = 1.19

Question 43

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.

A) E(X) = 71.75
B) E(X) = 79.50
C) E(X) = 73.50
D) E(X) = 77.25

Question 44

The odds against an event E occurring are 4 to 1. What is the probability of E not occurring?

A) 0.5
B) 0.25
C) 0.6
D) 0.8

Question 45

In European roulette the wheel is divided into 37 compartments numbered 1 through 36 and 0. (In American roulette there are 38 compartments numbered 1 through 36, 0, and 00.) Find the expected value of the winnings on a $3 bet placed on red in European roulette. Round your answer to the nearest cent.

A) $0.08%
B) - $0.08%
C) - $0.03%
D) $0.03%

Question 46

Maria sees the growth of her business for the upcoming year as being tied to the gross domestic product (GDP). She believes that her business will grow (or contract) at the rate of 5%, 4.5%, 3%, 0%, or - 0.5% per year if the GDP grows (or contracts) at the rate of between 2 and 2.5%, between 1.5 and 2%, between 1 and 1.5%, between 0 and 1%, and between - 1 and 0%, respectively. Maria has decided to assign a probability of 0.13, 0.24, 0.39, 0.19, and 0.05, respectively, to each outcome. At what rate does Maria expect her business to grow next year?

A) 2.615%
B) 3.005%
C) 2.875%
D) 2.325%

Question 47

A woman purchased a $10,000, 1-year term-life insurance policy for $150. Assuming that the probability that she will live another year is 0.993, find the company's expected gain.

A) E = $80
B) E = $100
C) E = $140
D) E = $90

Question 48

The odds in favor of an event E occurring are 12 to 8. What is the probability of E occuring?

A) 1.40
B) 0.40
C) 0.60
D) 0.67
E) 1.50

Question 49

Find the expected value of a random variable X having the following probability distribution:

A) E(X) = 0.79
B) E(X) = 1.07
C) E(X) = 0.73
D) E(X) = 0.86

Question 50

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.

A) E = 0.081
B) E = 0.156
C) E = 0.106
D) E = 0.141

Question 51

If a sample of 8 batteries is selected from a lot of 11, of which 4 are defective, what is the expected number of defective batteries? Round your answer to next whole number.

A) 7
B) 10
C) 12
D) 5
E) 6

Question 52

A panel of 50 economists was asked to predict the average prime interest rate for the upcoming year. The results of the survey follow: Based on this survey, what does the panel expect the average prime interest rate to be next year?

A) E = 5.154
B) E = 5.374
C) E = 4.104
D) E = 5.724

Question 53

In American roulette a player may bet on a split (two adjacent numbers). In this case, if the player bets $1 and either number comes up, the player wins $14 and gets his $1 back. If neither comes up, he loses his $1 bet. Find the expected value of the winnings on a $1 bet placed on a split. Round your answer to the nearest cent.

A) - $0.06%
B) - $0.21%
C) $0.06%
D) $0.21%

Question 54

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 two-child family

Question 55

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
The area of a histogram associated with a probability distribution is a number between 0 and 1.

Question 56

A panel of 64 economists was asked to predict the average unemployment rate for the upcoming year. The results of the survey follow: On the basis of this survey, what does the panel expect the average unemployment rate to be next year?

A) 2.82
B) 2.32
C) 2.68
D) 3.30
E) 3.39

Question 57

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

Question 58

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: What does the management expect the sales to be next year?

A) 32.34
B) 15.34
C) 28.98
D) 29.61
E) 18.7

Question 59

The management of MultiVision, a cable TV company, intends to submit a bid for the cable television rights in one of two cities, A or B. If the company obtains the rights to city A, the probability of which is 0.3, the estimated profit over the next 10 yr is $10 million; if the company obtains the rights to city B, the probability of which is 0.4, the estimated profit over the next 10 yr is $7 million. The cost of submitting a bid for rights in city A is $300,000 and that of city B is $300,000. ​
By comparing the expected profits for each venture, determine whether the company should bid for the rights in city A or city B.

A) City A
B) City B

Question 60

Find the expected value of a random variable X having the following probability distribution:

A) E(X) = 2.3125
B) E(X) = 1.0975
C) E(X) = 2.1875
D) E(X) = 2.3825

Question 61

If a player placed a $8 bet on red and a $5 bet on black in a single play in American roulette, what would be the expected value of his winnings? Round your answer to the nearest cent.

A) cents
B) cents
C) cents
D) cents
E) cents

Question 62

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. Give your answer to three decimal places, if necessary. E = __________

Question 63

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. A game between two persons is fair if the expected value to both persons is zero.

A) False. This is contrary to the definition.
B) True. This follows from the definition.

Question 64

Steffi feels that the odds in favor of her winning her tennis match tomorrow are 5 to 4. What is the (subjective) probability that she will win her match tomorrow? Give your answer to four decimal places, if necessary.
The probability is __________.

Question 65

Maria sees the growth of her business for the upcoming year as being tied to the gross domestic product (GDP). She believes that her business will grow (or contract) at the rate of 5%, 4.5%, 3%, 0%, or - 0.5% per year if the GDP grows (or contracts) at the rate of between 2 and 2.5%, between 1.5 and 2%, between 1 and 1.5%, between 0 and 1%, and between - 1 and 0%, respectively.
Maria has decided to assign a probability of 0.11, 0.23, 0.43, 0.19, and 0.04, respectively, to each outcome. At what rate does Maria expect her business to grow next year? Give your answer to three decimal places, if necessary.
__________ %

Question 66

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. Give your answer to the nearest cent. E (X) = $ __________

Question 67

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. Find the expected number of machines that will break down on a given day. Give your answer to two decimal places, if necessary. E = __________

Question 68

The odds against an event E occurring are 1 to 7. What is the probability of E not occurring? Give your answer to three decimal places, if necessary.
The probability is __________.

Question 69

In American roulette a player may bet on a split (two adjacent numbers). In this case, if the player bets $1 and either number comes up, the player wins $17 and gets his $1 back. If neither comes up, he loses his $1 bet. Find the expected value of the winnings on a $1 bet placed on a split. Round your answer to the nearest cent.
$ __________

Question 70

Roger Hunt intends to purchase one of two car dealerships currently for sale in a certain city. Records obtained from each of the two dealers reveal that their weekly volume of sales, with corresponding probabilities, are as follows. Dahl Motors Farthington Auto Sales The average profit/car at Dahl Motors is $367, and the average profit/car at Farthington Auto Sales is $439.
If Roger's objective is to purchase the dealership that generates the higher weekly profit, which dealership should he purchase? (Compare the expected weekly profit for each dealership.)

A) Farthington Auto Sales
B) Dahl Motors
C) either of the two

Question 71

A woman purchased a $10,000, 1-year term-life insurance policy for $150. Assuming that the probability that she will live another year is 0.991, find the company's expected gain.
E = $ __________

Question 72

A panel of 50 economists was asked to predict the average prime interest rate for the upcoming year. The results of the survey follow. Based on this survey, what does the panel expect the average prime interest rate to be next year? Give your answer to three decimal places, if necessary. E = __________

Question 73

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 videocassettes. How many videocassettes can a customer entering this store be expected to buy? Give your answer to two decimal places, if necessary. E = __________

Question 74

A buyer for Discount Fashions, an outlet for women's apparel, is considering buying a batch of clothing for $61,000. She estimates that the company will be able to sell it for $80,000, $75,000, or $70,000 with probabilities of 0.20, 0.40, and 0.40, respectively. Based on these estimates, what will be the company's expected gross profit?

A) $13,000
B) $23,400
C) $7,800
D) $15,600
E) $10,400

Question 75

In a lottery, 4,000 tickets are sold for $1 each. One first prize of $2,000, 1 second prize of $800, 3 third prizes of $120, and 10 consolation prizes of $12 are to be awarded. What are the expected net earnings of a person who buys one ticket?

A) cents
B) cents
C) cents
D) cents
E) cents

Question 76

In European roulette the wheel is divided into 37 compartments numbered 1 through 36 and 0. (In American roulette there are 38 compartments numbered 1 through 36, 0, and 00.) Find the expected value of the winnings on a $2 bet placed on black in European roulette. Round your answer to the nearest cent.
$ __________

Question 77

Find the expected value of a random variable X having the following probability distribution. Give your answer to two decimal places, if necessary. E (X) = __________

Question 78

Find the expected value of a random variable X having the following probability distribution. Give your answer to four decimal places, if necessary. E (X) = __________

Question 79

Bob, the proprietor of Midland Lumber, feels that the odds in favor of a business deal going through are 7 to 6. What is the (subjective) probability that this deal will not materialize?

A) 0.4647
B) 0.4622
C) 0.4615
D) 0.4460

Question 80

Steffi feels that the odds in favor of her winning her tennis match tomorrow are 2 to 10. What is the (subjective) probability that she will win her match tomorrow?

A) 0.1612
B) 0.1660
C) 0.1699
D) 0.1667