Deck 4: Probability and Probability Distributions

Question 1

There are two types of random variables: discrete and continuous.

Answer 1

True

Question 2

The number of homes sold by a real estate agency in a three week period is an example of a continuous random variable.

Answer 2

False

Question 3

The probability distribution of the number of accidents in Grand Rapids, Michigan, each day is given by Based on this distribution, the standard deviation of the number of accidents per day is approximately:
Round your answer to two decimal places.

A) 6.95
B) 2.33
C) 2.64
D) 1.53
E) 15

Answer 3

1.53

Question 4

A random variable is any variable the numerical value of which is determined by a random experiment, and, thus, by chance.

Question 5

The expected number of heads in 500 tosses of an unbiased coin is:

A) 150
B) 200
C) 250
D) 300
E) 500

Question 6

Which of following statements is true regarding the probability distribution for a discrete random variable X?

A) The probabilities must be nonnegative.
B) The probabilities must sum to 1.
C) The random variable must take on positive values between 0 and 1.
D) Both the probabilities must be nonnegative and the probabilities must sum to 1.
E) Both the probabilities must be nonnegative and the random variable must take on positive values between 0 and 1.

Question 7

The number of students living off-campus is an example of a discrete random variable.

Question 8

The weight of a box of candy bars is an example of a discrete random variable since there are only a specific number of bars in the box.

Question 9

The mean of a discrete random variable x:

A) is denoted by
B) is denoted by E(x), because it is the value of x one can expect to find on average by numerous repetitions of the random experiment that generates the variable's actual values
C) is correctly described by both "is denoted by" and "is denoted by E(x), because it is the value of x one can expect to find on average by numerous repetitions of the random experiment that generates the variable's actual values"
D) is always = 0
E) is none of these

Question 10

If the mean of the random variable x is larger than the mean of the random variable y, then the variance of x must be larger than the variance of y.

Question 11

A table, graph, or formula that associates each possible value of a discrete random variable, x with its probability of occurrence, p(x), is called a discrete probability distribution.

Question 12

In a set of all possible values a discrete random variable is countable, because the variable can assume values only at specific points on a scale of values, with inevitable gaps in-between.

Question 13

Which of the following correctly describes the nature of discrete quantitative variables?

A) They are characteristics possessed by persons or objects, called elementary units, in which we are interested.
B) They can assume values only at specific points on a scale of values, with inevitable gaps between successive observations
C) When dealing with such variables, we can count all possible observations and, with some exceptions, that count leads to a finite result.
D) Is correctly described by "they are characteristics possessed by persons or objects, called elementary units, in which we are interested" and "when dealing with such variables, we can count all possible observations and, with some exceptions, that count leads to a finite result."
E) All of these.

Question 14

The probability distribution for a discrete variable x is a formula, a table, or a graph providing p(x), the probability associated with each of the values of x.

Question 15

The time required to assemble a computer is an example of a discrete random variable.

Question 16

If x is a discrete random variable, then x can take on only one of two possible values.

Question 17

The graph of a discrete random variable looks like a histogram where the probability of each possible outcome is represented by a bar.

Question 18

The probability distribution of the number of accidents in Grand Rapids, Michigan, each day is given by This distribution is an example of:

A) continuous probability distribution
B) discrete probability distribution
C) conditional probability distribution
D) an expected value distribution
E) Is not a probability distribution.

Question 19

A table, formula, or graph that shows all possible values x a random variable can assume, together with their associated probabilities P(x) is called:

A) a discrete probability distribution
B) a continuous probability distribution
C) a bivariate probability distribution
D) a law of total probability
E) all of these

Question 20

The probability distribution of the number of accidents in Grand Rapids, Michigan, each day is given by Based on this distribution, the expected number of accidents in a given day is:

A) 4.62
B) 1.47
C) 2.15
D) 1.81
E) 1

Question 21

A false negative in screening tests is the event that the test is positive for a given condition, given that the person does not have the condition.

Question 22

A false positive in screening (e.g., home pregnancy tests) represents the event that:

A) the test is negative for a given condition, given that the person does not have the condition
B) the test is positive for a given condition, given that the person does not have the condition
C) the test is negative for a given condition, given that the person has the condition
D) the test is positive for a given condition, given that the person has the condition

Question 23

Let x be a random variable with the following distribution:
Round your answer to three decimal places, if necessary.
a. Find the expected value of x.
______________
b. Find the standard deviation of x.
______________
c. What is the probability that x is further than one standard deviation from the mean?
______________

Question 24

A false positive in screening tests is the event that the test is negative for a given condition, given that the person has the condition.

Question 25

From experience, a shipping company knows that the cost of delivering a small package within 24 hours is $16.20. The company charges $16.95 for shipment but guarantees to refund the charge if delivery is not made within 24 hours. If the company fails to deliver only 3% of its packages within the 24-hour period, what is the expected gain per package?
______________

Question 26

A false negative in screening tests (e.g., Steroid testing of athletes) represents the event that:

A) the test is negative for a given condition, given that the person does not have the condition
B) the test is positive for a given condition, given that the person does not have the condition
C) the test is negative for a given condition, given that the person has the condition
D) the test is positive for a given condition, given that the person has the condition

Question 27

Which of the following statements is false?

A) The mean of a discrete probability distribution is equal to the square root of the variance.
B) The expected value of a discrete probability distribution is the long-run average value if the experiment is to be repeated many times.
C) The standard deviation of a discrete probability distribution measures the average variation of the random variable from the mean.
D) All of the above.
E) None of these.

Question 28

Approximately 5% of the bolts coming off a production line have serious defects. Two bolts are randomly selected for inspection. Find the probability distribution for x, the number of defective bolts in the sample.
a. P(x = 0) = P(both bolts are not defective)
______________
b. P(x = 1) = P(one bolt is defective)
______________
c. P(x = 2) = P(both bolts are defective)
______________
d. Find E(x).
______________
e. Find
.
______________

Question 29

Steve takes either a bus or the subway to go to work with probabilities 0.25 and 0.75, respectively. When he takes the bus, he is late 40% of the days. When he takes the subway, he is late 30% of the days. If Steve is late for work on a particular day, what is the probability that he took the bus?
______________

Question 30

The random variable x is defined as the number of mistakes made by a typist on a randomly chosen page of a physics thesis. The probability distribution follows.
Round your answer to four decimal places, if necessary.
a. What is E(x)?
______________
b. Find
.
______________
c. Find P(x < 1).
______________
d. Find
.
______________
e. In what fraction of pages in the thesis would the number of mistakes made be within two standard deviations of the mean?
______________

Question 31

Medical case histories indicate that different illnesses may produce identical symptom. Suppose a particular set of symptoms, which we will denote as event H, occurs only when any one of these illnesses: A, B, or C occurs. (For the sake of simplicity, we will assume that illnesses A, B, and C are mutually exclusive.) Studies show these probabilities of getting the three illnesses: The probabilities of developing the symptoms H, given a specific illness, are: Assuming that an ill person shows the symptoms H, what is the probability that the person has illness A?
Round your answer to four decimal places.
______________

Question 32

Bayes' Rule is a formula for revising an initial subjective (prior) probability value on the basis of results obtained by an empirical investigation and for, thus, obtaining a new (posterior) probability value.

Question 33

In the "Quick Draw" casino card game, a player chooses one card from a deck of 52 well-shuffled cards. If the card the player selects is the king of hearts, the player wins $104; if the card is an ace, the player wins $78; if the card selected is anything else, the player loses $13. Considering a negative amount won to be a loss, how much should the player expect to win in one play of "Quick Draw"?
$ ______________

Question 34

Screening tests (e.g., AIDS testing) are evaluated on the probability of a false negative or a false positive, and both of these are:

A) conditional probabilities
B) probability of the intersection of two events
C) probability of the union of two events
D) marginal probabilities
E) probability of dependent events

Question 35

The neighborhood deli specializes in New York Style Cheesecake. The cheesecakes are made fresh daily and any unsold cheesecake is donated to the Salvation Army soup kitchen to feed the homeless. Each cheesecake costs $5 to make and sells for $11. The daily demand for cheesecake (the number the deli could sell if they had a cheesecake for everyone who wanted one) has the following distribution.

a. What is the expected demand for cheesecake?
______________
b. What is the expected daily profit if the deli makes only one cheesecake per day?
$ ______________
c. What is the expected daily profit if the deli makes only two cheesecakes per day?
$ ______________

Question 36

A sample is selected from one of two populations, S₁ and S₂, with probabilities P(S₁) = 0.80 and P(S₂) = 0.20. If the sample has been selected from S₁, the probability of observing an event A is P(A / S₁) = 0.15. Similarly, If the sample has been selected from S₂, the probability of observing A is P(A / S₂) = 0.25.
Round your answer to four decimal places, if necessary
a. If a sample is randomly selected from one of the two populations, what is the probability that event A occurs?
______________
If a sample is randomly selected and event A is observed, what is the probability that the sample was selected:
b. From population S₁?
______________
c. From population S₂?
______________

Question 37

Let the random variable x represent the number of cars owned by a family. Assume that x can take on five values: 0, 1, 2, 3, 4. A partial probability distribution is shown below:

a. Find the probability that a family owns three cars.
______________
b. Find the probability that a family owns between 1 and 3 cars, inclusive.
______________
c. Find the probability that a family owns no more than one car.
______________
d. What is the probability that a family owns more than two cars?
______________
e. What is the probability that a family owns at most 3 cars?
______________

Question 38

The probability distribution of your winnings at a casino's card game is shown below.

a. What is the chance you win more than $1 if you play just once?
______________
b. How much should you expect to win if you play the game once?
______________
c. After breaking the bank at the casino playing this card game, you decide to open your own casino where the customers can play your favorite card game. How much should you charge the players if you want to have an average profit of $1 per play?
______________

Question 39

Let x denote the weight gain in pounds per month for a calf. The probability distribution of x is shown below.

a. Find the average weight gain in pounds per month for a calf.
______________
b. Find the variance of the weight gain.
______________
c. What is
?
______________
d. What is
?
______________
e. What is
?
______________

Question 40

Given a set of events that are mutually exclusive and exhaustive and an event A, the law of total probability states that P(A) can be expressed as .

Question 41

If P(A) = 0, P(B) = 0.4, and P(A B) = 0, then events A and B are independent.

Question 42

If P(A) = 0.4, P(B) = 0.5, and P(A B) = 0.7, then P(A B) = 0.2.

Question 43

The complement of an event A, denoted by , consists of all the simple events in the sample space S that are not in A.

Question 44

If P(A) = 0.60, P(B) = 0.40, and P(B / A) = 0.60, then P(A / B) = 0.24.

Question 45

Suppose A and B are mutually exclusive events where P(A) = 0.2 and P(B) = 0.3. Then P(A B) = 0.5.

Question 46

The intersection of events A and B is the event that A or B or both occur.

Question 47

Two events A and B are said to be independent if and only if P(A / B) = P(B) or P(B / A) = P(A).

Question 48

If P(A) > 0, P(B) > 0, and P(A B) = 0, then the events A and B are independent.

Question 49

If P(A) = 0.4, P(B) = 0.5, and P(A B) = 0.20, then the events A and B are mutually exclusive.

Question 50

If P(A) = 0.3, P(A B) = 0.7, and P(A B) = 0.2, then P(B) = 0.2.

Question 51

If A and B are independent events with P(A) = 0.30 and P(B) = 0.50, then P(A/B) is 0.15.

Question 52

The conditional probability of event B, given that event A has occurred is defined by: .

Question 53

If A and B are two independent events with P(A) = 0.25 and P(B) = 0.45, then P(A B) = 0.70.

Question 54

The probability that event A will not occur is 1- .

Question 55

Conditional probability is the probability that an event will occur, with no other events taken into consideration.

Question 56

Two events A and B are said to be independent if P(A B) = P(A) + P(B).

Question 57

Suppose A and B are mutually exclusive events where P(A) = 0.1 and P(B) = 0.7, then P(A B) = 0.8.

Question 58

If P(A) > 0 and P(B) > 0, then when A and B are mutually exclusive events, they are also dependent events.

Question 59

Two events A and B are said to mutually exclusive if P(A B) = 0.

Question 60

If P(A / B) = P(A), then events A and B are said to be independent.

Question 61

The additive rule of probability is used to compute the probability for an intersection of two or more events: namely, given two events A and B, and also

Question 62

Which of the following best describes the concept of marginal probability?

A) It is a measure of the likelihood that a particular event will occur, regardless of whether another event occurs.
B) It is a measure of the likelihood that a particular event will occur, given the fact that another event has already occurred or is certain to occur.
C) It is a measure of the likelihood of the simultaneous occurrence of two or more events.
D) It is a direct way for defining the sample space of an experiment.
E) It is a measure of the likelihood that a particular event will not occur.

Question 63

Which one of the following is always true for two events, A and B?

A) If A and B are independent, they are also mutually exclusive.
B) If A and B are dependent, they are also mutually exclusive.
C) If P(A / B) = P(AB), A and B are independent.
D) If A and B are mutually exclusive, then AB can never occur on the same trial of an experiment.
E) All of these.

Question 64

A missile designed to destroy enemy satellites has a 0.80 chance of destroying its target. If the government tests three missiles by firing them at a target, what is the probability all three fail to destroy the target? (Assume the missiles perform independently.)
______________

Question 65

All the outcomes contained in one or the other of two events (or possibly in both) constitute the union of two events.

Question 66

If P(A) = 0.40, P(B) = 0.30 and P(A B) = 0.12, then A and B are:

A) dependent events
B) independent events
C) mutually exclusive events
D) disjoint events
E) none of these

Question 67

Suppose A and B are events where P(A) = 0.4, P(B) = 0.5, and P(A B) = 0.2. Then P(B / A) = 0.4.

Question 68

All the outcomes (simple events) contained in one or the other of two random events, or possibly in both, make up:

A) the events of an experiment
B) the intersection of two events
C) the probability space of an experiment
D) the union of two events
E) the complement of the other event

Question 69

Studies have shown a particular television commercial is understood by 25% of first grade students and 80% of fourth grade students. If a television advertising agency randomly selects one first grader and one fourth grader, what is the probability neither child would understand the commercial, assuming the children's reactions are independent?
______________

Question 70

If P(A) = 0.42 and P(B) = 0.38, then P(A B) is:

A) 0.80
B) 0.58
C) 0.04
D) 0.42
E) Cannot be determined from the given information.

Question 71

If P(A) = 0.30, P(B) = 0.40 and P(A B) = 0.20, then P(A / B) is:

A) 0.12
B) 0.08
C) 0.67
D) 0.50
E) 0.3

Question 72

The probability of an event and the probability of its complement always sum to:

A) -1
B) 0
C) 1
D) any value between 0 and 1
E) 0.5

Question 73

If events A and B are mutually exclusive, then the probability of both events occurring simultaneously is equal to:

A) -1
B) 0
C) 1
D) any value between 0 and 1
E) 0.5

Question 74

Two events A and B are said to be dependent if and only if:

A) P(A) = P(B)
B) P(A) increases along with P(B)
C) P(A) increases as P(B) decreases
D) event A is affected or changed by the occurrence of event B
E) P(A) < P(B)

Question 75

Suppose P(A) = 0.4, P(B) = 0.3, and P(A B) = 0. Which one of the following statements correctly defines the relationship between events A and B?

A) Events A and B are independent, but not mutually exclusive.
B) Events A and B are mutually exclusive, but not independent.
C) Events A and B are neither mutually exclusive nor independent.
D) Events A and B are both mutually exclusive and independent.
E) None of these.

Question 76

Which of the following clearly describes the general multiplicative rule of probability?

A) It is a rule of probability theory that is used to compute the probability for the occurrence of a union of two or more events: for any two events, A and B,.
B) It is a rule of probability theory that is used to compute the probability for the occurrence of a union of two or more events: for any two events A and B,.
C) It is a rule of probability theory that is used to compute the probability for an intersection of two or more events: for any two events, A and B,
D) It is a rule of probability theory that is used to compute the probability for an intersection of two or more events: for any two events A and B,
E) All of these.

Question 77

If P(A) = 0.80, P(B) = 0.70 and P(A B) = 0.90, then P(A B) is:

A) 0.60
B) 0.56
C) 0.72
D) 0.63
E) 0.5

Question 78

An election is being held to fill two "at large" city council seats. Two Republicans and three Democrats are running for office. Assume the candidates are equally likely to be elected, and independent of each other.
a. What are the possible outcomes of the election?
______________
b. What is the probability both seats are filled by Republicans?
______________
c. What is the probability one Republican and one Democrat are elected to the two city council seats?
______________

Question 79

If P(A/B) = P(A), or P(B/A) = P(B), then events A and B are said to be:

A) mutually exclusive
B) disjoint
C) independent
D) dependent
E) B contains A.

Question 80

The addition law of probability theory is used to compute the probability for the occurrence of a union of two or more events; namely, given two events A and B,