Deck 9: Counting, Probability Distributions, and Further Topics in Probability

Question 1

Prepare a probability distribution for the experiment. Let $x$ represent the random variable, and let $P$ represent theprobability.

-T hree coins are tossed, and the number of tails is noted.

A)
B)
C)
D)

Answer 1

Question 2

Prepare a probability distribution for the experiment. Let $x$ represent the random variable, and let $P$ represent theprobability.

-Three cards are drawn from a deck. The number of kings is counted.

A)
B)
C)
D)

Answer 2

Question 3

Prepare a probability distribution for the experiment. Let $x$ represent the random variable, and let $P$ represent theprobability.

-Two balls are drawn from a bag in which there are 4 red balls and 2 blue balls. The number of blue balls is counted.

A)
B)
C)
D)

Answer 3

Question 4

Prepare a probability distribution for the experiment. Let $x$ represent the random variable, and let $P$ represent theprobability.

-Four cards are drawn from a deck. The number of red tens is counted.

A)
B)
C)
D)

Question 5

Prepare a probability distribution for the experiment. Let $x$ represent the random variable, and let $P$ represent theprobability.

-A ball player batting .300 comes to bat 3 times in a game. The number of hits is counted.

A)
B)
C)
D)

Question 6

Prepare a probability distribution for the experiment. Let $x$ represent the random variable, and let $P$ represent theprobability.

-Four coins are tossed and the number of heads is counted.

A)
B)
C)
D)

Question 7

Prepare a probability distribution for the experiment. Let $x$ represent the random variable, and let $P$ represent theprobability.

-Three balls are drawn from a bag containing 5 red and 3 green balls. The number of green balls is counted.

A)
B)
C)
D)

Question 8

Prepare a probability distribution for the experiment. Let $x$ represent the random variable, and let $P$ represent theprobability.

-Five cards are drawn from a deck. The number of tens is counted.

A)
B)
C)
D)

Question 9

Prepare a probability distribution for the experiment. Let $x$ represent the random variable, and let $P$ represent theprobability.

-A field goal kicker has a kicking average of .75 and he tries 3 field goals in a game. The number of field goals is counted.

A)
B)
C)
D)

Question 10

Give the probability distribution and sketch the histogram.

-A class of 44 students took a 10 -point quiz. The frequency of scores is given in the table.

A)
B)
C)
D)

Question 11

Give the probability distribution and sketch the histogram.

-At a safety program for hunters, each member of a class of 25 took 5 shots at a target. The number of bullseyes is shown in the table.

A)
B)
C)
D)

Question 12

Give the probability distribution and sketch the histogram.

-The telephone company kept track of the calls for the correct time during a 24 -hour period for two weeks. The results are shown in the table.

A)
B)
C)
D)

Question 13

Give the probability distribution and sketch the histogram.

-At a supermarket during the 6 p.m. rush, the manager counted the number of customers waiting in each of the 12 checkout lines. The results are shown in the table.

A)
B)
C)
D)

Question 14

Give the probability distribution and sketch the histogram.

-Five rats are inoculated against a disease. After an incubation period, the number contracting the disease is noted. The experiment is repeated 20 times, with the results shown in the table.

A)
B)
C)
D)

Question 15

Find the expected value of the random variable in the experiment.

-Three coins are tossed, and the number of tails is noted.

A)2
B) 1
C) 1.5
D) 1.75

Question 16

Find the expected value of the random variable in the experiment.

-Three cards are drawn from a deck without replacement. The number of aces is counted.

A) .2308
B) .2174
C) 1.0134
D) 1

Question 17

Find the expected value of the random variable in the experiment.

-A bag contains six marbles, of which four are red and two are blue. Suppose two marbles are chosen at random and $\mathrm{X}$ represents the number of red marbles in the sample.

A) 1.4
B) 1.33
C) .933
D) 1

Question 18

Find the expected value of the random variable in the experiment.

-Five rats are inoculated against a disease. The number contracting the disease is noted and the experiment is repeated 20 times. Find the probability distribution and give the expected number of rats contracting the disease.

A) 2.3
B) 2.4
C) .9
D) 1

Question 19

Find the expected value for the random variable.

-

A) 4
B) 3.7
C) 3.3
D) 3.5

Question 20

Find the expected value for the random variable.

-

A) 7.58
B) 7.98
C) 7.62
D) 9

Question 21

Find the expected value for the random variable.

-

A) 6.51
B) 8.34
C) 9.18
D) 9

Question 22

Find the expected value for the random variable.

-

A) 28.58
B) 28
C) 22.68
D) 30.14

Question 23

Find the expected value for the random variable.

-A business bureau gets complaints as shown in the following table. Find the expected number of complaints per day.

A) 3.01
B) 2.98
C) 2.73
D) 2.85

Question 24

Find the expected value for the random variable.

-For a certain animal species, the probability that a female will have a certain number of offspring in a given year is given in the table below. Find the expected number of offspring per year.

A) 1.58
B) 2
C) 1.75
D) 1.38

Question 25

Find the expected value for the random variable.

-A business bureau gets complaints as shown in the following table. Find the expected number of complaints per day.

A) 2.98
B) 2.85
C) 3.01
D) 2.73

Question 26

Find the expected value for the random variable x having this probability function.

- $$\begin{array}{ll}\mathrm{a}=5 & \mathrm{~b}=6 \\\mathrm{c}=7 & \mathrm{~d}=8\end{array}$$

A) 5.2
B) 7
C) 6.5
D) 5.5

Question 27

Find the expected value for the random variable x having this probability function.

- $$\begin{array}{lll} \mathbf{a}=10 & \mathbf{b}=12 & \mathbf{c}=14 \\\mathbf{d}=16 & \mathbf{e}=18 &\end{array}$$

A) 17.2
B) 14.4
C) 15.8
D) 14

Question 28

Find the expected value for the random variable x having this probability function.

- $$\begin{array}{lll}\mathbf{a}=17 & \mathbf{b}=18 & \mathbf{c}=19 \\\mathbf{d}=20 & \mathbf{e}=21 &\end{array}$$

A) 15.6
B) 14.8
C) 15.4
D) 19

Question 29

Find the expected value for the random variable x having this probability function.

- $$\mathbf{a}=90 \quad \mathbf{b}=95$$
$$\mathbf{c}=100$$

A) 95
B) 95.5
C) 85.5
D) 104.5

Question 30

Solve the problem.

-Suppose you pay $\$ 1.00$ to roll a fair die with the understanding that you will get back $\$ 3.00$ for rolling 6 or 4 . What are your expected winnings?

A) $\$ 0$
B) $\$ 3.00$
C) $\$ 1.00$
D) $-\$ 2.00$

Question 31

Solve the problem.

-Suppose you pay $\$ 1.00$ to roll a fair die with the understanding that you will get back $\$ 3$ for rolling a 1 or a 5 , nothing otherwise. What are your expected winnings?

A) $\$ 0$
B) $\$ 1.00$
C) $\$ 3$
D) $-\$ 1.00$

Question 32

Solve the problem.

-A contractor is considering a sale that promises a profit of $\$ 25,000$ with a probability of .7 or a loss (due to bad weather, strikes, and such) of $\$ 17,000$ with a probability of .3 . What is the expected profit?

A) $\$ 17,500$
B) $\$ 29,400$
C) $\$ 8000$
D) $\$ 12,400$

Question 33

Solve the problem.

-Experience shows that a ski lodge will be full (174 guests) if there is a heavy snow fall in December, while only partially full (59 guests) with a light snow fall. What is the expected number of guests if the probability for a heavy snow fall is .40 ?

A) 104.4
B) 105
C) 69.6
D) 128

Question 34

Solve the problem.

-An insurance company has written 89 policies of $\$ 50,000,450$ of $\$ 25,000$ , and 900 of $\$ 10,000$ on people of age 20. If the probability that a person will die at age 20 is .001, how much can the company expect to pay during the year the policies were written?

A) $\$ 0$
B) $\$ 2470$
C) $\$ 247,000$
D) $\$ 24,700$

Question 35

Solve the problem.

-An insurance company says that at age 50 one must choose to take $\$ 10,000$ at age $60, \$ 30,000$ at 70 , or $\$ 50,000$ at 80 ( $\$ 0$ death benefit). The probability of living from 50 to 60 is .83 , from 50 to 70 , .61, and from 50 to $80,0.47$ . Find the expected value at each age.

A) 60 - $8300
70 - $14,100
80 - $23,500
B) 60 - $6100
70 - $18,300
80 - $23,500
C) 60 - $8300
70 - $24,900
80 - $41,500
D) 60 - $8300
70 - $18,300
80 - $23,500

Question 36

Solve the problem.

-At age 50, Ann must choose between taking $\$ 19,000$ at age 60 if she is alive then, or $\$ 28,000$ at age 70 if she is alive then. The probability for a person aged 50 living to be 60 and 70 is .88 and . 63 , respectively. Using expected value, what is Ann's best option?

A) $\$ 19,000$ at age 60
B) $\$ 28,000$ at age 70

Question 37

Solve the problem.

-If 2 cards are drawn from a deck of 52 cards, what is the expected number of spades?

A).75
B) .50
C) .47
D) .25

Question 38

Solve the problem.

-Find the expected number of boys in a family of 4 children.

A) 2
B) 2.5
C) 3
D) 2.75

Question 39

Solve the problem.

-Find the expected number of boys in a family of 4 children.

A) 2.5
B) 3
C) 2.75
D) 2

Question 40

Solve the problem.

-A male decreases his life expectancy by 1 year on average, for every point that his blood pressure is above 120. The average life expectancy is 76 years. Find the life expectancy for a male whose blood pressure is 124 .

A) 62
B) 76
C) 72
D) 73

Question 41

Evaluate the expression.

-9 !

A) 362,880
B) 362,871
C) 362,889
D) 40,320

Question 42

Evaluate the expression.

- $5^{\mathrm{P}} 4$

A) 24
B) 120
C) 5
D) 1

Question 43

Evaluate the expression.

- $8^{\mathrm{P}_{4}}$

A) 1680
B) 4
C) 70
D) 2

Question 44

Evaluate the expression.

- $10^{P_{2}}$

A) 8
B) 45
C) 90
D) 19

Question 45

Evaluate the expression.

- ${ }_{6} \mathrm{P}_{5}$

A) 21
B) 1
C) 0
D) 720

Question 46

Evaluate the expression.

- ${ }_{12} \mathrm{C}_{3}$

A) 220
B) 440
C) 362,880
D) 4

Question 47

Evaluate the expression.

- ${ }_{12} C_{0}$

A) $39,916,800$
B) 12
C) 1
D) 11

Question 48

Evaluate the expression.

- ${ }_{38} \mathrm{C}_{1}$

A) 39
B) 1
C) 38
D) 37

Question 49

Evaluate the expression.

- ${ }_{23} \mathrm{C}_{14}$

A) 817,215
B) 817,190
C) 817,205
D) 817,162

Question 50

Use the multiplication principle to solve the problem.

-How many different 5 -digit sequences can be formed using the digits $0,1, \ldots, 9$ if repetition of digits is allowed?

A) 59,049
B) 100,000
C) 6
D) 45

Question 51

Use the multiplication principle to solve the problem.

-License plates are made using 3 letters followed by 2 digits. How many plates can be made if repetition of letters and digits is allowed?

A) $11,881,376$
B) 175,760
C) $1,757,600$
D) 100,000

Question 52

Use the multiplication principle to solve the problem.

-If 11 newborn babies are randomly selected, how many different gender sequences are possible?

A) 121
B) 22
C) $39,916,800$
D) 2048

Question 53

Use the multiplication principle to solve the problem.

-A restaurant offers 6 possible appetizers, 10 possible main courses, and 4 possible desserts. How many different meals are possible at this restaurant?
(Two meals are considered different unless all three courses are the same).

A) 240
B) 216
C) 20
D) 230

Question 54

Use the multiplication principle to solve the problem.

-How many different 4 -letter radio station call letters can be made if repeats are allowed and the first letter must be $\mathrm{K}$ .

A) 1000
B) 78
C) 17,576
D) 456,976

Question 55

Use the multiplication principle to solve the problem.

-How many different 7 -digit phone numbers are possible if the first digit cannot be a 0 ?

A) $1,000,000$
B) $10,000,000$
C) $9,000,000$
D) 900,000

Question 56

Use the multiplication principle to solve the problem.

-A biologist is attempting to classify plant species by assigning 6 initials to each species. How many classifications are possible using this method?

A) $1,000,000$
B) $60,466,176$
C) $308,915,776$
D) 156

Question 57

Use the multiplication principle to solve the problem.

-Mark can remember only the first 5 digits of his friend's phone number. He also knows that the number has 7 digits and that the last digit is not a 0 . If Mark were to dial all of the possible numbers and if it takes him 16 seconds to try each one, how long would it take to try every possibility?

A) 26.7 minutes
B) 24 minutes
C) 5.3 minutes
D) 2.7 minutes

Question 58

Solve the problem.

-A musician plans to perform 10 selections. In how many ways can she arrange the musical selections?

A) $39,916,800$
B) 100
C) 10
D) $3,628,800$

Question 59

Solve the problem.

-A pollster wants to minimize the effect the order of the questions has on a person's response to a survey. How many different surveys are required to cover all possible arrangements if there are 8 questions on the survey?

A) 5040
B) 40,320
C) 8
D) 64

Question 60

Solve the problem.

-There are 10 members on a board of directors. If they must elect a chairperson, a secretary, and a treasurer, how many different slates of candidates are possible?

A) 1000
B) 720
C) 120
D) $3,628,800$

Question 61

Solve the problem.

-In a game of musical chairs, 9 children will sit in 8 chairs arranged in a row (one child will not find a chair). In how many ways can 8 of the children find seats?

A) 40,320
B) 72
C) 363,020
D) 362,880

Question 62

Solve the problem.

-In how many ways can 2 letters be chosen from the set $\{A, B, C, D, E, F\}$ if order is important and no repeats are allowed?

A) 15
B) 30
C) 12
D) 36

Question 63

Solve the problem.

-A classical music concert is to consist of 5 cello pieces, 3 choral works, and 2 pieces for piano. In how many ways can the program be arranged if a piano piece must come first?

A) 362,880
B) 725,760
C) $3,628,800$
D) 80,640

Question 64

Solve the problem.

-In a lottery, players must pick 6 different numbers from 1 to 22 . How many ways are there to choose the 6 numbers if order is important?

A) 74,613
B) $53,721,360$
C) $113,379,904$
D) 132

Question 65

Solve the problem.

-From 10 names on a ballot, 3 will be elected to a political party committee. In how many ways can the committee of 3 be formed if each person will have a different responsibility?

A) $3,628,800$
B) 4320
C) 1000
D) 720

Question 66

Solve the problem.

-The library is to be given 8 books as a gift. The books will be selected from a list of 20 titles. If each book selected must have a different title, how many possible selections are there?

A) 5079
B) 125,970
C) 6034
D) 40,320

Question 67

Solve the problem.

-In a certain lottery, 4 numbers between 1 and 10 inclusive are drawn. These are the winning numbers. How many different selections are possible? Assume that the order in which the numbers are drawn is not important.

A) 5040
B) 10,000
C) 210
D) 24

Question 68

Solve the problem.

-How many ways can an IRS auditor select 6 of 12 tax returns for an audit?

A) 924
B) 720
C) 665,280
D) $2,985,984$

Question 69

Solve the problem.

-Five cards are drawn at random from an ordinary deck of 52 cards. In how many ways is it possible to draw all black cards?

A) 131,560 ways
B) 32,890 ways
C) 65,780 ways
D) 263,120 ways

Question 70

Solve the problem.

-Five cards are drawn at random from an ordinary deck of 52 cards. In how many ways is it possible to draw two red cards and three black cards?

A) 422,500 ways
B) $1,690,000$ ways
C) 845,000 ways
D) $1,267,500$ ways

Question 71

Solve the problem.

-Five cards are drawn at random from an ordinary deck of 52 cards. In how many ways is it possible to draw two red aces and two black jacks?

A) 192 ways
B) 144 ways
C) 48 ways
D) 1,152 ways

Question 72

Solve the problem.

-From a group of 10 women and 15 men, a researcher wants to randomly select 7 women and 7 men for a study. In how many ways can the study group be selected?

A) 772,200
B) $4,457,400$
C) $33,037,200$
D) 6555

Question 73

Solve the problem.

-Three student representatives are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if two must be male and one female?

A) 2
B) 12
C) 4
D) 6

Question 74

Solve the problem.

-Three student representatives, a president, a secretary, and a treasurer, are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if the president must be a woman and the secretary and treasurer must be men?

A) 12
B) 4
C) 3
D) 6

Question 75

Solve the problem.

-Two student representatives, a treasurer and a secretary, are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if the two must not be the same sex?

A) 10
B) 6
C) 3
D) 12

Question 76

Solve the problem.

-Awards are to be presented to seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul. How many different orders are possible for the awards if Maria and Olivia must be next to each other?

A) 1220
B) 1440
C) 1680
D) 720

Question 77

Solve the problem.

-Awards are to be presented to seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul. How many different orders are possible for the awards if Karen is to receive the first award and Lyle the last?

A) 120
B) 24
C) 840
D) 360

Question 78

Solve the problem.

-Awards are to be presented to seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul. How many different orders are possible for the awards if the men are to receive their awards first, and then the women?

A) 5040
B) 2
C) 144
D) 72

Question 79

Solve the problem.

-A bag contains 3 blue, 4 red, and 5 green marbles. Four marbles are drawn at random from the bag. How many different samples are possible which include exactly two red marbles?

A) 18
B) 168
C) 6
D) 672

Question 80

Solve the problem.

-A bag contains 4 blue, 3 red, and 5 green marbles. Four marbles are drawn at random from the bag. How many different samples are possible which include at least one marble of each color?

A) 90
B) 270
C) 60
D) 540