Deck 8: Probability: the Mathematics of Chance

Question 1

Suppose a trial consists of rolling two dice and reporting the sum of the numbers rolled. What are the possible outcomes? Are they equally likely?

Answer 1

Possible outcomes are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12. They are not equally likely.

Question 2

A pizza can be made with any of the following toppings: cheese, pepperoni, sausage, mushrooms, anchovies, green peppers, or olives. How many different three-topping pizzas can be made? Doubling of any topping is not allowed.

Answer 2

35

Question 3

We wish to make a spinner that will be numbered 1 to 4, but will have the probability of spinning a "1" be 0.5. Draw the face for such a spinner.

Answer 3

Answers may vary. Any spinner face which shows half the face labeled "1" is correct.

Question 4

A pair of dice is rolled. Sam says there are 36 outcomes in the sample space for this procedure and Sally says there are 11 outcomes in the sample space. Explain how they could both be correct.

Question 5

A spinner with regions numbered 1 to 4 is spun and a coin is tossed. Both the region number and heads or tails is recorded. Write the sample space.

Question 6

A pizza can be made with any of the following toppings: cheese, pepperoni, sausage, mushrooms, anchovies, green peppers, or olives. Doubling of any topping is not allowed. What is the probability that a randomly created three-topping pizza will contain mushrooms?

Question 7

Three coins are flipped simultaneously and it is recorded whether each coin lands heads or tails. List the sample space.

Question 8

Suppose a game has four outcomes: A, B, C, and D. The probability of outcome A is 0.4, the probabilities of each of the other outcomes is 0.2. A player receives $2 if outcome A occurs, $3 if outcome B occurs, $1 if outcome C occurs, and must pay $5 if outcome D occurs. What is the mean value of one trial of this game?

Question 9

You roll two dice and record the sum shown. What is the probability of the event "the sum is greater than 2"?

Question 10

If the probability that Kerry gets an A in English class is 0.82, what is the probability that Kerry does not get an A?

Question 11

A student is taking a five-question true/false test. If the student chooses answers at random, what is the probability of getting all questions correct?

Question 12

Below is a probability model for the number of automobiles owned by a randomly chosen family in a large town. What is the mean number of automobiles owned?

Question 13

Suppose a trial consists of rolling a single die and reporting the number that is rolled. What are the possible outcomes? Are they equally likely?

Question 14

A license plate code consists of two letters followed by three digits. The letters cannot repeat, but the digits can. What is the probability that a randomly chosen plate has all three digits the same?

Question 15

The mean value of one trial of a carnival game is −$.05. Explain what this means.

Question 16

We must create a license plate code that consists of two letters followed by three digits. The letters cannot repeat, but the digits may. How many such license plate codes can be formed?

Question 17

Exactly one of three contestants will win a game show. The probability that Terry wins is 0.25 and the probability that Chris wins is 0.65. What is the probability that Toni, the remaining contestant wins?

Question 18

A pair of fair dice is rolled and the sum of the faces showing is recorded. What is the probability of getting a sum greater than 9?

Question 19

Find the probability of drawing a three or a heart from a regular (bridge) deck of cards. (Such a deck consists of four suits of 13 cards each. The suits are hearts, spades, diamonds, and clubs. The cards are 1 through 10, Jack, Queen, and King.)

Question 20

Suppose three fair coins are tossed and the number of heads that appear is recorded. What is the probability of getting exactly two heads?

Question 21

For a sample space {2,3,4,5,6}, two events are defined as follows:
Event A: A chosen number is greater than 4
Event B: A chosen number is a prime number; that is, 2, 3, or 5
Are those events A and B dependent or independent?

Question 22

Suppose a trial consists of rolling two dice and reporting the smaller of the two numbers rolled. What are the possible outcomes? Are they equally likely?

Question 23

A game consists of tossing a coin and rolling a six-sided die. The results can be recorded easily; for example, if heads shows on the coin and a 4 shows on the die, you can record this as H4. List the sample space for the results of this game.

Question 24

How many possible ways are there for drawing three spades from a regular (bridge) deck of cards?

Question 25

A spinner numbered 1 through 10 is spun and one die is tossed simultaneously. The number spun and the number rolled are recorded. How many outcomes are in the sample space?

A) 60
B) 16
C) 10
D) 6

Question 26

What is the probability of drawing three cards that are all spades from a regular (bridge) deck of cards?

Question 27

Suppose you use a spinner to choose a random number between 0 and 1. Which is more likely-choosing a number between 1/3 and 2/3 or between 1/2 and 3/4?

Question 28

Three dice are tossed. The number rolled on each die is recorded. How many outcomes are in the sample space?

A) 18
B) 72
C) 216
D) 42

Question 29

A continuous probability model has a constant density curve on the range 0 to 12. What is the height of the density curve?

Question 30

A spinner numbered 1 through 10 is spun and one die is rolled simultaneously. The sum of the number spun and the number rolled is recorded. How many outcomes are in the sample space?

A) 60
B) 16
C) 15
D) 10

Question 31

A game consists of tossing three coins. Find the probability model for the number of coins showing heads.

Question 32

For a sample space {1,2,3,4,5,6,7,8}, two events are defined as follows:
Event A: A chosen number is greater than 4
Event B: A chosen number is a prime number; that is, 2, 3, 5, or 7
Are those events A and B dependent or independent?

Question 33

A die is rolled and a coin is flipped simultaneously. The number rolled on the die and whether the coin lands heads or tails is recorded. How many outcomes are in the sample space?

A) 8
B) 6
C) 10
D) 12

Question 34

According to the central limit theorem, how does the standard deviation of averages over four observations compare with the standard deviation of individual observations?

Question 35

Two coins are flipped at the same time and heads or tails is recorded. The sample space for this is:

A) {H, T}.
B) {HH, HT, TT}.
C) {HH, HT, TH, TT}.
D) {H, H, T, T}.

Question 36

A football stadium snack bar conducted a survey and found that 30 customers bought one item, 52 customers bought two items, 21 customers bought three items, and 12 customers bought four items. What is the mean number of items purchased per customer? Round your answer to the nearest tenth.

Question 37

The mean weight of a collection of potatoes in a shipment to a fruit market is 1.3 pounds, with a standard deviation of 0.35 pounds. The distribution of weights is approximately Normal. What is the probability that one potato chosen at random will weigh more than 1.65 pounds?

Question 38

In the manufacturing process for ball bearings, the mean diameter is 5 mm with a standard deviation of 0.002 mm. Every hour a sample of 20 bearings is drawn, measured, and the mean diameter of the sample found. What is the standard deviation _x of the sample mean?

Question 39

The mean weight of a collection of potatoes in a shipment to a fruit market is 1.3 pounds, with a standard deviation of 0.35 pounds. The distribution of weights is approximately Normal. What fraction of the potatoes in the shipment will weigh between 0.25 and 2.35 pounds?

Question 40

In the manufacturing process for ball bearings, the mean diameter is 5 mm with a standard deviation of 0.002 mm. Between what two measurements will 95% of all diameters of ball bearings be found?

Question 41

Two coins are flipped and the number that landed on heads is recorded. The outcomes in the sample space are all equally likely.

Question 42

Use the following to answer
You roll two dice and record the sum shown. The sample space is the set of all possible sums, which are 2, 3, . . . , 11, 12.
What is the complement of the event "the sum is less than 6 or greater than 7"?

A) The outcome is 6.
B) The outcome is 6 or 7.
C) The outcome is less than 7.
D) The outcome is 6 or larger.

Question 43

Use the following to answer
You roll two dice and record the sum shown. The sample space is the set of all possible sums, which are 2, 3, . . . , 11, 12.
The probability of the event "the sum is 5" is:

A) 1/6.
B) 1/8.
C) 1/9.
D) 1/12.

Question 44

If you roll two dice and record their sum, the events "even" and "less than or equal to 6" are:

A) disjoint.
B) independent.
C) Both A and B is correct.
D) Neither A nor B is correct.

Question 45

If you randomly select a committee of two from a group of two men and two women, what is the probability that the group is all female?

A) 1/2
B) 1/4
C) 1/6
D) 1/12

Question 46

To function correctly, each of two parts of a circuit must work. If an event denotes that the circuit functions correctly, what is the complement of the event?

A) Exactly one of two parts is defective.
B) At least one of two parts is defective.
C)
D) Both parts are defective.
None of parts is defective.

Question 47

If a fair die is rolled once, what is the probability of getting a number less than five?

A) 1/6
B) 1/2
C) 2/3
D) 5/6

Question 48

Three dice are rolled and the number rolled on each die is recorded. The outcomes in the sample space are all equally likely.

Question 49

Use the following to answer
You roll two dice and record the sum shown. The sample space is the set of all possible sums, which are 2, 3, . . . , 11, 12.
Are the 11 outcomes equally likely?

A) Yes
B) No

Question 50

Three dice are tossed. The sum of the numbers rolled is recorded. How many outcomes are in the sample space?

A) 18
B) 12
C) 16
D) 72

Question 51

A fair coin is tossed three times. Find the probability of getting exactly two heads.

A) 1/2
B) 1/3
C) 2/3
D) 3/8

Question 52

Use the following to answer
You roll two dice and record the sum shown. The sample space is the set of all possible sums, which are 2, 3, . . . , 11, 12.
What other outcome has a probability equal to the probability of "the sum is 2"?

A) The sum is greater than 2.
B) The sum is less than 10.
C) The sum is greater than 11.
D) None of the above.

Question 53

You use a random number generator that produces the digits 1 through 9 with equal probability. You draw four digits and make them into a number; for example, 0, 2, 4, 4 would be the number 0244 or just 244. What is the probability of getting a number strictly greater than 1000?

A) 1/10
B) 1001/10,000
C) 1000/9999
D) None of the above.

Question 54

Two fair dice are rolled and the sum rolled is recorded. Find the probability that the sum is 4.

A) 1/3
B) 1/12
C) 4/11
D) 1/9

Question 55

A computer is programmed to randomly print two letters in a row without repeating a letter. What is the probability that the first combination printed is the word "DO"?

A) 1/650
B) 1/325
C) 2/325
D) 1/26

Question 56

Three dice are rolled and the sum of the numbers rolled is recorded. The outcomes in the sample space are all equally likely.

Question 57

There are seven blue and six black socks in a drawer. One is pulled out at random. Find the probability that it is black.

A) 6/13
B) 6/7
C) 1/2
D) 1/6

Question 58

We need to create three-digit code numbers that must begin with a 7. How many such codes can be made?

A) 700
B) 107
C) 100
D) 27

Question 59

Two coins are flipped and heads or tails is recorded. The outcomes in the sample space are all equally likely.

Question 60

Two dice are rolled and the sum is recorded. What sis the most likely outcome?

A) 7
B) 8
C) 9
D) 12

Question 61

For a casino game, you pay $4 and roll a die. The casino keeps your $4 but pays you back the number of dollars you roll. (Roll a 6 and you get $6.) Your average loss per bet is:

A) $0.
B) $0.50.
C) $1.
D) $4.

Question 62

If two events A and B have probabilities as P(A) = 0.45 and P(B) = 0.7, then what is the smallest possible value for P(A and

A) 0.1
B) 0.15
B)?
C) 0.3
D) 0.35

Question 63

A fair die is rolled. If a number 1 or 2 appears, you will receive $5. If any other number appears, you will pay $2. What is the mean value of one trial of this game?

A) $1/3
B) $3
C) $3/2
D) −$3

Question 64

Use the following information for Questions
A random number generator is modeled by a uniform probability model with a density curve that has height 1/10 between 0 and 10 and height 0 everywhere else.
For this model, what is

A) 0
B) 1/2
C) 1
D) None of the above.

Question 65

Use the following information for Questions
A random number generator is modeled by a uniform probability model with a density curve that has height 1/10 between 0 and 10 and height 0 everywhere else.
For this model, what is

A) 0
B) 1/2
C) 1
D) None of the above .

Question 66

At a certain discount store, the number of people in checkout lines varies. The probability model for the number of people in a randomly chosen line is: What is the mean number of people in a line?

A) 2.5
B) 15.92
C) 2.81
D) 2.89

Question 67

Suppose a game has three outcomes, A, B, and C with probabilities P(A) = 0.2, P(B) = 0.3, and P(C) = 0.5. A player will receive $3 when outcome A occurs, $4 when outcome B occurs, and will have to pay $2 when outcome C occurs. What is the mean value of one trial of the game?

A) $0.80
B) $1.66
C) $1.00
D) $5.00

Question 68

The Olympic flag consists of five intertwined circles, one in each of the colors black, blue, green, red, and yellow. What is the probability that a random coloring of the five circles using these colors will produce the exact match of the Olympic symbol?

A) 1/5
B) 1/3125
C) 5/120
D) 1/120

Question 69

A continuous probability model has the following density curve: What is

A) 1/8
B) 1/4
C) 1/2
D) None of the above.

Question 70

We need to create serial numbers that start with one of the letters a, b, c, d, or f followed by three nonrepeating digits. How many serial numbers can be created?

A) 5000
B) 2520
C) 725
D) 3600

Question 71

The marketing department of an electronics manufacturer has done research on the number of television sets owned by families in a large town. The probability model for the number of sets owned by a randomly chosen family is given below. What is the mean number of sets per family?

A) 0.96
B) 1.5
C) 1.77
D) 1.73

Question 72

A sample space consists of three outcomes, X, Y, and Z. Which of the following could be a legitimate assignment of probabilities to the outcomes?

A)
B)
C)
D)

Question 73

Use the following information for Questions
A random number generator is modeled by a uniform probability model with a density curve that has height 1/10 between 0 and 10 and height 0 everywhere else.
For this model, what is

A) 0
B) 1/2
C) 1
D) None of the above.

Question 74

A computer system requires users to have an access code that consists of a three-digit number that is not allowed to start with zero and cannot repeat digits. How many such codes are possible?

A) 990
B) 648
C) 729
D) 720

Question 75

A raffle ticket costs $5. First-prize and second-prize winners will be drawn at random. The probability of winning the $100 first prize is 1/40, and the probability of winning the $25 second prize is 1/20. What is the mean winnings for one play, taking into account the $5 cost of the ticket?

A) −$1.25
B) −$0.875
C) $3.375
D) $3.75

Question 76

We need to create code words that use three letters of the alphabet. Repeating of letters is allowed. How many code words can be created?

A) 17,576
B) 15,600
C) 78
D) 75

Question 77

Exactly one of Terry, Chris, and Kim will attend a party. The probability Terry attends is 0.31 and the probability Chris attends is 0.5. What is the probability that Kim attends?

A) 0.33
B) 0.5
C) 0.81
D) 0.19

Question 78

Use the following information for Questions
A random number generator is modeled by a uniform probability model with a density curve that has height 1/10 between 0 and 10 and height 0 everywhere else.
For this model, what is

A) 0
B) 1/2
C) 1
D) None of the above.

Question 79

A sample space contains three outcomes, A, B, and C. Which of the following could be a legitimate assignment of probabilities to the outcomes?

A)
B)
C)
D)

Question 80

If there is a 0.8 probability of rain today, what is the probability it will not rain?

A) 0.8
B) 0.5
C) 0.2
D) 0.1