Deck 13: The Nature of Probability

Question 1

All the cows in a certain herd are white-faced. The probability that a white-faced calf will be born by mating with a certain bull is 0.8. Suppose 2 cows are bred to the same bull. Find the probability, that no white-faced calves will be born.
​
Please round your answer to four decimal places.
​
P = __________

Answer 1

0.0400

Question 2

Find the binomial probability.
​ ​
Round your answer to five significant digits.

Answer 2

Question 3

Suppose you are taking a true-false test with ten questions. If you guess at the answers on this test, find the probability of getting fewer than nine correct answers. ​

A) 0.9944
B) 0.9928
C) 0.9868
D) 0.9935
E) 0.9893

Answer 3

0.9893

Question 4

Suppose that research has shown that the probability that a missile penetrates enemy defenses and reaches its target is 0.1. Find the smallest number of identical missiles that are necessary in order to be 90% certain of hitting the target at least once. ​

A) 27
B) 22
C) 20
D) 21
E) 19

Question 5

Suppose the National League team has a probability of of winning a World Series game and the American League team has a probability of . The series is over as soon as one team wins four games. Find the probability that the series is over in seven games. Please round your answer to four decimal places.
​
P = __________

Question 6

Find the binomial probability. ​ ​

A)
B)
C)
D)
E)

Question 7

Suppose you are taking a true-false test with ten questions. If you guess at the answers on this test, find the probability of getting fewer than three correct answers. Please round your answer to four decimal places.
​
P = __________

Question 8

Find the probability of obtaining exactly four threes on six rolls of a fair die. Please round your answer to three decimal places.
​
P = __________

Question 9

Lymnozyme cures most infections in Koi fish caused by bacteria; in fact, it has been shown to be 98% effective if used according to the directions. If 14 Koi with a bacterial infection are treated, what is the probability that 13 of the fish are cured? Please round your answer to four decimal places.
​
P = __________

Question 10

Find the probability of obtaining exactly two threes on four rolls of a fair die. ​

A) 0.559
B) 0.243
C) 0.068
D) 0.116
E) 0.131

Question 11

A researcher chooses three leaves from a target environment and classifies each sample as fungus-free or contaminated. Suppose that a leaf has a probability of 0.4 of being infected. Find the probability, that two leaves infected. ​

A) 0.490
B) 0.181
C) 0.314
D) 0.144
E) 0.288

Question 12

Lymnozyme cures most infections in Koi fish caused by bacteria; in fact, it has been shown to be 98% effective if used according to the directions. If 11 Koi with a bacterial infection are treated, what is the probability that 10 of the fish are cured? ​

A) 0.1788
B) 0.1798
C) 0.1973
D) 0.9481
E) 0.1767

Question 13

Lymnozyme cures most infections in Koi fish caused by bacteria; in fact, it has been shown to be 91% effective if used according to the directions. If 5 Koi with a bacterial infection are treated, what is the probability that 4 of the fish are cured? ​

A) 0.3086
B) 0.3261
C) 0.8837
D) 0.3076
E) 0.3055

Question 14

In a certain office, seven men determine who will pay for coffee by each flipping a coin. If one of them has an outcome that is different from the other six, he must pay for the coffee. What is the probability that in any play of the game there will be an "odd man out"? Please round your answer to four decimal places.
​
P = __________

Question 15

Suppose that research has shown that the probability that a missile penetrates enemy defenses and reaches its target is 0.2. Find the smallest number of identical missiles that are necessary in order to be 50% certain of hitting the target at least once.
​
__________ missiles

Question 16

A researcher chooses three leaves from a target environment and classifies each sample as fungus-free or contaminated. Suppose that a leaf has a probability of 0.4 of being infected. Find the probability, that three leaves are infected. Please round your answer to three decimal places.
​
P = __________

Question 17

Lymnozyme cures most infections in Koi fish caused by bacteria; in fact, it has been shown to be 96% effective if used according to the directions. If 7 Koi with a bacterial infection are treated, what is the probability that 6 of the fish are cured? Please round your answer to four decimal places.
​
P = __________

Question 18

In a certain office, six men determine who will pay for coffee by each flipping a coin. If one of them has an outcome that is different from the other five, he must pay for the coffee. What is the probability that in any play of the game there will be an "odd man out"? ​

A) 0.1875
B) 0.1125
C) 0.0188
D) 0.1313
E) 0.0750

Question 19

Suppose the National League team has a probability of of winning a World Series game and the American League team has a probability of . The series is over as soon as one team wins four games. Find the probability that the series is over in seven games. ​

A) 0.1518
B) 0.2718
C) 0.1318
D) 0.1438
E) 0.1294

Question 20

All the cows in a certain herd are white-faced. The probability that a white-faced calf will be born by mating with a certain bull is 0.7. Suppose 6 cows are bred to the same bull. Find the probability, that no white-faced calves will be born. ​

A) 0.0004
B) 0.0008
C) 0.0023
D) 0.0007
E) 0.0051

Question 21

Suppose that in an assortment of 20 calculators, there are 5 with defective switches. Draw with and without replacement. If two machines are selected at random, what is the probability that both have defective switches?
​
probability with replacement = __________
​
probability without replacement = __________

Question 22

Suppose a slot machine has three independent wheels as shown in the figure.
​
Find the probability P(3 bar). Give your answer to four decimal places, if required.
​

Question 23

What is the probability that two people in your class (assume a class of 30 students) have the same birthday? ​

A) 70.6%
B) 73.7%
C) 0.1%
D) 65.2%
E) 29.4%

Question 24

Find the requested probability. ​ if ​

A)
B)
C) ​
D) ​
E) ​

Question 25

A game consists of removing a card from a deck of 52 cards. If the card is a face card, you win $3 and the game is over. If it is not a face card, remove another card. If that one is a face card, you win $3. Repeat the process again, and then again (for a total of 4 times). If you have not removed any face card, then you lose the game, and must pay $3. Should you play this game? ​

A) no
B) yes

Question 26

A game consists of removing a card from a deck of 52 cards. If the card is a face card, you win $2 and the game is over. If it is not a face card, remove another card. If that one is a face card, you win $2. Repeat the process again, and then again (for a total of 4 times). If you have not removed any face card, then you lose the game, and must pay $2. Should you play this game?
​
Answer yes or no.

Question 27

Suppose a slot machine has three independent wheels as shown in the figure. ​
Find the probability P(3 plum). Round your answer to four decimal places if necessary.
​ ​

A) 0
B) 0.05
C) 0.0045
D) 0.002
E) 0.01

Question 28

Suppose events A, B, and C are independent and , , . ​
Find the probability of .
​

A)
B)
C)
D)
E)

Question 29

Suppose events A, B, and C are independent and ​ ​
Find the probability .
​

A)
B)
C)
D)
E)

Question 30

Suppose events A, B, and C are independent and , , .
​
Find the probability of .

Question 31

Suppose a die is rolled twice and let
​
A = {first toss is a prime}
B = {first toss is 2}
C = {second toss is a 3}
D = {second toss is 2}
​
Find the probability .

Question 32

Suppose that in an assortment of 6 calculators, there are 3 with defective switches. Draw with and without replacement. If two machines are selected at random, what is the probability that both have defective switches? ​

A) with : , without :
B) both :
C) with : , without :
D) both :
E) none of these

Question 33

What is the probability of obtaining at least one head when a coin is flipped six times?

Question 34

Suppose events A, B, and C are independent and ​ ​
Find the probability .
​

A)
B)
C)
D)
E)

Question 35

What is the probability that two people in your class (assume a class of 30 students) have the same birthday? Please, express your answer to 0.1%.
​
__________%

Question 36

What is the probability of obtaining at least one head when a coin is flipped four times? ​

A)
B)
C)
D)
E)

Question 37

Which of the following is more probable?
A. Flipping a coin 4 times and obtaining at least 3 heads.
B. Flipping a coin 5 times and obtaining at least 3 heads.

A) A
B) B
C) same
D) cannot tell

Question 38

Suppose events A, B, and C are independent and
​ ​
Find the probability .

Question 39

Suppose a die is rolled twice and let ​
A = {first toss is a prime}
B = {first toss is 5}
C = {second toss is a 6}
D = {second toss is 5}
​
Find the probability .
​

A)
B)
C)
D)
E)

Question 40

Suppose events A, B, and C are independent and
​ ​
Find the probability .

Question 41

A sorority has 12 members, 4 of whom are full members and 8 are pledges. Two persons are selected at random from the membership list of the sorority. Find the probability: the second person selected is a pledge. ​

A)
B)
C)
D)
E)

Question 42

The probability of drawing a club from a deck of cards is ; what are the odds in favor of drawing a club? ​

A) 1 to 4
B) 3 to 1
C) 1 to 5
D) 2 to 3
E) 1 to 3

Question 43

A single card is drawn from a standard deck of cards. Find the probabilities if the given information is known about the chosen card. A face card is a jack, queen, or king. ​ ​

A) ​0
B)
C) ​1
D)
E)

Question 44

A single card is drawn from a standard deck of cards. Find the probabilities if the given information is known about the chosen card. A face card is a jack, queen, or king.
​

Question 45

A sorority has 36 members, 28 of whom are full members and 8 are pledges. Two persons are selected at random from the membership list of the sorority. Find the probability: the second person selected is a pledge.

Question 46

Two cards are drawn from a deck of cards. Find the requested probability.
​
The second card drawn is a club if the first card drawn was a club.

Question 47

Which of the following is more probable? Answer A, B, or same.

A) Flipping a coin 3 times and obtaining at least 2 heads.
B) Flipping a coin 4 times and obtaining at least 2 heads.

Question 48

A sorority has 20 members, 12 of whom are full members and 8 are pledges. Two persons are selected at random from the membership list of the sorority. Find the probability: the first person selected is a pledge.

Question 49

Two cards are drawn from a standard deck of cards, and one of the two cards is noted and removed. Find the probability of drawing the second card, given the information about the removed card.
​

Question 50

Two cards are drawn from a standard deck of cards, and one of the two cards is noted and removed. Find the probability of drawing the second card, given the information about the removed card. ​ ​

A)
B)
C)
D)
E)

Question 51

Find the requested probability. Give your answer to two decimal places, if required.
​ if ​ __________

Question 52

A sorority has 27 members, 21 of whom are full members and 6 are pledges. Two persons are selected at random from the membership list of the sorority. Find the probability: the first person selected is a pledge. ​

A)
B)
C)
D)
E)

Question 53

Choose a natural number between 1 and 100, inclusive. What is the probability that the number chosen is not a multiple of 4?

Question 54

Use estimation to select the best response. Do not calculate. ​
The probability of correctly guessing a telephone number is about
​

A) 1 out of 100
B) 1 out of 10
C) 1 out of 1,000,000
D) 1 out of 10,000
E) 1 out of 1,000

Question 55

Suppose a pair of dice are rolled. Consider the sum of the numbers on the top of the dice and find the probability.
​
7, given that at least one die came up 2

Question 56

Suppose that you roll two dice. You will be paid $5 if you roll a double. You will not receive anything for any other outcome. How much should you be willing to pay for the privilege of rolling the dice? ​

A) $0.69
B) $0.14
C) $0.28
D) $0.83
E) $5.00

Question 57

The probability of drawing a diamond from a deck of cards is ; what are the odds in favor of drawing a diamond?
​
The odds are __________ to __________.

Question 58

Choose a natural number between 1 and 100, inclusive. What is the probability that the number chosen is not a multiple of 5? ​

A)
B)
C)
D)
E)

Question 59

Two cards are drawn from a deck of cards. Find the requested probability. ​
The second card drawn is a spade if the first card drawn was a spade.
​

A)
B)
C)
D)
E)

Question 60

Suppose a pair of dice are rolled. Consider the sum of the numbers on the top of the dice and find the probability. ​
8, given that at least one die came up 3
​

A)
B)
C)
D)
E)

Question 61

An oil-drilling company knows that it costs $65,000 to sink a test well. If oil is hit, the income for the drilling company will be $550,000. If only natural gas is hit, the income will be $428,373. If nothing is hit, there will be no income. If the probability of hitting oil is 1/65 and if the probability of hitting gas is 1/35, what is the expectation for the drilling company? Should the company sink the test well? ​

A) - $55,216; should dig
B) $44,299; should dig
C) - $44,299; should dig
D) - $44,299; should not dig
E) - $42,695; should not dig

Question 62

What is the expectation for the $1 five-number bet on a U.S. roulette wheel?
​ ​
$__________

Question 63

A box contains one each of $2, $8, $40, $65, and $100 bills. You reach in and withdraw one bill. What is the expected value? Please round your answer to the nearest dollar.
​
$__________

Question 64

Use estimation to find the expected value. Do not calculate.
​
The expected value of playing a $1 game of blackjack is $0.04, find the netted value after playing the game 100 times.
​
The netted value is $__________.

Question 65

A box contains one each of $4, $8, $25, $90, and $100 bills. You reach in and withdraw one bill. What is the expected value? ​

A) $114
B) $45
C) $25
D) $20
E) $227

Question 66

In a TV game show, four prizes are hidden on a game board which contains 60 spaces. One prize is worth $1,500, two prizes are worth $750, and the other prize is worth $50. The remaining spaces contain no prizes. The game show host offers a sure prize of $50 not to play this game. Should the contestant choose the sure prize or play the game?
​
Answer play, doesn't matter, or don't play.

Question 67

Suppose that you roll two dice. You will be paid $20 if you roll a double. You will not receive anything for any other outcome. How much should you be willing to pay for the privilege of rolling the dice? Please round your answer to the nearest cent.
​
$__________

Question 68

Heights (in inches) obtained by a group of people in a random survey is reported in the table. What is the expected height (in inches)? Heights
Probability
55
0)007
60
0)024
65
0)112
70
0)357
75
0)357
80
0)112
85
0)024
90
0)007
​

A) 80 in.
B) 78.3 in.
C) 78 in.
D) 71.8 in.
E) 72.5 in.

Question 69

In a TV game show, four prizes are hidden on a game board which contains 80 spaces. One prize is worth $4,000, two prizes are worth $2,000, and the other prize is worth $100. The remaining spaces contain no prizes. The game show host offers a sure prize of $100 not to play this game. Should the contestant choose the sure prize or play the game? ​

A) doesn't matter
B) play
C) don't play

Question 70

What is the expectation for the $1 four-number bet on a U.S. roulette wheel? ​ ​

A) $0.05
B) $0.08
C) - $0.01
D) - $0.08
E) - $0.05

Question 71

Consider a state lottery that has a weekly television show. On this show, a contestant receives the opportunity to win $1 million. The contestant picks from four hidden windows. Behind each is one of the following: $250,000, $125,000, $1 million, or a "stopper." Before beginning, the contestant is offered $300,000 to stop. Mathematically speaking, should the contestant take the $300,000?
​
Answer yes, no, or doesn't matter.

Question 72

What is the expectation for the $1 black bet on a U.S. roulette wheel?
​ ​
$__________

Question 73

Krinkles potato chips is having a "Lucky Seven Sweepstakes." The one grand prize is $70,000; 7 second prizes each pay $7,000; 77 third prizes each pay $700; and 777 fourth prizes each pay $70. What is the expectation of this contest, if there are 6 million entries? Please round your answer to the nearest cent.
​
$__________

Question 74

What is the expectation for the $1 black bet on a U.S. roulette wheel? ​ ​

A) $0.05
B) - $0.01
C) $0.02
D) - $0.08
E) - $0.05

Question 75

Krinkles potato chips is having a "Lucky Seven Sweepstakes." The one grand prize is $70,000; 7 second prizes each pay $7,000; 77 third prizes each pay $700; and 777 fourth prizes each pay $70. What is the expectation of this contest, if there are 9 million entries? ​

A) $0.05
B) $0.86
C) $0.03
D) $0.25
E) $0.01

Question 76

A single card is selected from an ordinary deck of cards. The sample space is shown in the figure below. Find the probability. ​ ​ ​

A)
B) ​
C) ​
D) ​
E) ​

Question 77

Consider a state lottery that has a weekly television show. On this show, a contestant receives the opportunity to win $1 million. The contestant picks from four hidden windows. Behind each is one of the following: $225,000, $300,000, $1 million, or a "stopper." Before beginning, the contestant is offered $400,000 to stop. Mathematically speaking, should the contestant take the $400,000? ​

A) yes
B) I don't know
C) doesn't matter
D) no

Question 78

Last year, 1,303 calculators were returned to the manufacturer. If 81,000 were produced, assign a number to specify the probability that a particular calculator would be returned. ​

A) about 0.016
B) about 0.130
C) about 0.033
D) about 0.058
E) about 0.012

Question 79

An oil-drilling company knows that it costs $95,000 to sink a test well. If oil is hit, the income for the drilling company will be $875,000. If only natural gas is hit, the income will be $340,648. If nothing is hit, there will be no income. If the probability of hitting oil is and if the probability of hitting gas is , what is the expectation for the drilling company? Should the company sink the test well? Please round your answer to the nearest dollar.
​
The expectation is __________, the company __________ (should, shouldn't) dig.

Question 80

For the spinners assume that the pointer can never lie on a border line. Find . ​ ​

A)
B) ​
C) ​
D) ​
E) ​