Question 1

Which of the following sets of outcomes described below in I and II represent mutually exclusive events? I. "Your final course grade is an A"; "Your final course grade is a B."
II) "Your final course grade is an A"; "Your final course grade is a Pass."
A) Neither I nor II represent mutually exclusive events.
B) Both I and II represent mutually exclusive events.
C) Only I represents mutually exclusive events.
D) Only II represents mutually exclusive events.

Accepted Answer

In set I, it is impossible for a student to receive both an A and a B as their final grade, making them mutually exclusive events. However, in set II, it is possible for a student to receive an A and also receive a Pass, making them not mutually exclusive events.

Question 2

The intersection of events A = {apple pie, peach pie, pumpkin pie} and B = {cherry pie, blueberry pie, pumpkin pie} is ________.
A) {pumpkin pie}
B) {apple pie, peach pie, cherry pie, blueberry pie}
C) {apple pie, peach pie, pumpkin pie, cherry pie, blueberry pie}
D) {apple pie, peach pie, pumpkin pie, cherry pie, blueberry pie, pumpkin pie}

Accepted Answer

The intersection of two events consists of the elements that are in both events. Therefore, the only item that appears in both A and B is pumpkin pie.

Question 3

The intersection of events A and B, denoted by A ∩ B, ________.
A) contains outcomes that are either in A or B
B) contains outcomes that are both in A and B
C) does not contain outcomes that are either in A or B
D) does not contain outcomes that are both in A and B

Accepted Answer

The intersection of events A and B contains only the outcomes that are common to both A and B. Therefore, it contains outcomes that are both in A and B.

Question 4

An experiment consists of tossing three fair coins. What is the probability of tossing two tails?
A) 1/8
B) 1/4
C) 3/8
D) 1/2

Accepted Answer

Possible outcomes of tossing three coins are:
HHH, HHT, HTH, THH, HTT, THT, TTH, TTT
Out of these, three outcomes have two tails - HTT, THT, and TTH.
Therefore, probability of tossing two tails = (number of outcomes with two tails) / (total number of possible outcomes) = 3/8.

Question 5

Which of the following sets of outcomes described below in I and II represent mutually exclusive events? I. "You are a freshman"; "You are a senior."
II) "You are a freshman"; "You are an underclassman."
A) Neither I nor II represent mutually exclusive events.
B) Both I and II represent mutually exclusive events.
C) Only I represents mutually exclusive events.
D) Only II represents mutually exclusive events.

Accepted Answer

Mutually exclusive events cannot happen at the same time. In I, you cannot be both a freshman and a senior simultaneously, making them mutually exclusive. In II, being a freshman means you are an underclassman, so these events are not mutually exclusive.

Question 6

Mutually exclusive and collectively exhaustive events ________.
A) contain all outcomes in a sample space and may share common outcomes
B) contain all outcomes in a sample space and do not share common outcomes
C) do not have to contain all outcomes in a sample space but do not share common outcomes
D) do not have to contain all outcomes in a sample space and may share common outcomes

Accepted Answer

Mutually exclusive events are events that cannot occur at the same time, meaning they do not share any common outcomes. Collectively exhaustive events are events that cover all possible outcomes in a sample space. Therefore, mutually exclusive and collectively exhaustive events contain all outcomes in a sample space and do not share common outcomes.

Question 7

The odds for encountering rain on a 500-mile car trip are three to one. What is the probability of rain on this trip?
A) 0.25
B) 0.33
C) 0.50
D) 0.75

Accepted Answer

The odds of 3 to 1 for rain mean there are 3 chances for rain and 1 against. To find the probability, you divide the number of favorable outcomes (3) by the total number of outcomes (3+1=4), which gives 3/4 or 0.75.

Question 8

An analyst has a limit order outstanding on a stock. He argues that the probability that the order will execute before the close of trading is 0.20. Thus, the odds for the order executing before the close of trading are
A) one in four.
B) one in five.
C) four to one.
D) five to one.

Accepted Answer

The odds are the ratio of the probability of an event occurring to the probability of it not occurring. In this case, the probability of the order executing before the close of trading is 0.20, and the probability of it not executing is 1 - 0.20 = 0.80. Therefore, the odds are 0.20 / 0.80 = 1 / 4, which is one in four.

Question 9

The complement of an event A, within the sample space S, is the event consisting of ________.
A) all outcomes in A that are in S
B) all outcomes in S that are in A
C) all outcomes in S that are not in A
D) all outcomes in A that are not in S

Accepted Answer

The complement of an event A within the sample space S is the set of all outcomes in S that are not in A. This is denoted by A' or Ac. In other words, if an outcome is not in event A, then it must be in the complement A' within the same sample space S. Therefore, choice C is the correct answer.

Question 10

An analyst believes the probability that U.S. stock returns exceed long-term corporate bond returns over a five-year period is based on personal assessment. This type of probability is best characterized as a(n) ________.
A) a classical probability
B) empirical probability
C) objective probability
D) subjective probability

Accepted Answer

The probability in this case is based on the analyst's personal assessment, making it a subjective probability.

Question 11

A probability based on logical analysis rather than on observation or personal judgment is best referred to as a(n) ________.
A) classical probability
B) empirical probability
C) subjective probability
D) None of the above

Accepted Answer

A probability based on logical analysis rather than on observation or personal judgment is known as classical probability. It is based on the assumption that all events that make up a sample space are equally likely to occur, making it purely theoretical in nature.

Question 12

The union of events A and B, denoted by A ∪ B, ________.
A) contains all outcomes that are in A or B
B) contains all outcomes of an experiment
C) contains no outcomes that are in A and B
D) consists only of outcomes that are in A and B

Accepted Answer

The union of events A and B includes all outcomes that are in either A or B or both, so option A is the correct choice. Option B is incorrect because the union is not necessarily all outcomes of the experiment as some outcomes may not be in either A or B. Option C is incorrect because the union may contain outcomes that are in both A and B. Option D is incorrect because the union may include outcomes that are not exclusively in A and B.

Question 13

After extensive research, an analyst asserts that there is an 80% chance that ABC Corporation will beat its EPS forecast. Analogously, the odds for the company beating its EPS forecast are ________.
A) one in four
B) 1 in 1.25
C) four to one
D) 1.25 to 1

Accepted Answer

The odds of an event happening are calculated as the probability of the event happening divided by the probability of the event not happening. With an 80% chance (or 0.8 probability) of beating its EPS forecast, the probability of not beating the forecast is 20% (or 0.2). Thus, the odds are 0.8 / 0.2 = 4 to 1.

Question 14

For the sample space S = {apple pie, cherry pie, peach pie, pumpkin pie}, what is the complement of A = {pumpkin pie, cherry pie}?
A) {apple pie}
B) {peach pie}
C) {apple pie, peach pie}
D) {pumpkin pie, cherry pie}

Accepted Answer

The complement of a set A contains all the elements that are not in A. Therefore, in this case, the complement of A would be all the pies that are not pumpkin pie or cherry pie. The only pies that are not in set A are apple pie and peach pie. So, the complement of A is C = {apple pie, peach pie}.

Question 15

Let P(A) = 0.6, P(B) = 0.3, and P(A ∪ B^C) = 0.1. Calculate P(A ∩B). A) 0 B) 0.3 C) 0.9 D) Not enough information to calculate.

Accepted Answer

P(A ∪ B^C) = P(A) + P(B^C) - P(A ∩ B^C). Given P(A ∪ B^C) = 0.1 and P(A) = 0.6, we can find P(B^C) = 1 - P(B) = 0.7. Substituting these values, 0.1 = 0.6 + 0.7 - P(A ∩ B^C), solving gives P(A ∩ B^C) = 1.2 - 0.1 = 1.1, which is not possible. Thus, P(A ∩ B) must be 0.

Question 16

The odds against winning $1.00 in the lottery are 19 to 1. What is the probability of winning $1.00 in the lottery?
A) 0.05
B) 0.0526
C) 0.90
D) 0.95

Accepted Answer

The odds against winning are 19 to 1, which means that for every 20 tickets sold, 19 will lose and 1 will win. So the probability of winning is 1/20, which is 0.05.

Question 17

Which of the following represents a subjective probability?
A) The probability of rolling a 2 on a single die is one in six.
B) Based on a conducted experiment, the probability of tossing a head on an unfair coin is 0.6.
C) A skier believes she has a 10% chance of winning a gold medal.
D) Based on past observation, a manager believes there is a three-in-five chance of retaining an employee for at least one year.

Accepted Answer

Subjective probability is based on personal judgment or opinion rather than on precise calculation or empirical evidence. Choice C, where a skier believes she has a 10% chance of winning a gold medal, is an example of subjective probability because it is based on the skier's personal belief rather than objective data.

Question 18

The union of events A = {apple pie, peach pie, pumpkin pie} and B = {cherry pie, blueberry pie, pumpkin pie} is ________.
A) {pumpkin pie}
B) {apple pie, peach pie, cherry pie, blueberry pie}
C) {apple pie, peach pie, pumpkin pie, cherry pie, blueberry pie}
D) {apple pie, peach pie, pumpkin pie, cherry pie, blueberry pie, pumpkin pie}

Accepted Answer

The union of two sets includes all the elements from both sets, without repeating any element. In this case, the element "pumpkin pie" appears in both sets, but it is only included once in the union. Therefore, the union of A and B is {apple pie, peach pie, pumpkin pie, cherry pie, blueberry pie}.

Question 19

Assume the sample space S = {win, loss}. Select the choice that fulfills the requirements of the definition of probability.
A) P({win}) = 0.7, P({loss}) = 0.2
B) P({win}) = 0.7, P({loss}) = −0.3
C) P({win}) = 1.0, P({loss}) = 0.1
D) P({win}) = 0.5, P({loss}) = −0.5

Accepted Answer

The answer of Assume the sample space S = {win,...

Question 20

Which of the following represents an empirical probability?
A) The probability of tossing a head on a coin is 0.5.
B) The probability of rolling a 2 on a single die is one in six.
C) A skier believes she has a 0.10 chance of winning a gold medal.
D) Based on past observation, a manager believes there is a three-in-five chance of retaining an employee for at least one year.

Accepted Answer

Empirical probability is based on observed data rather than theoretical calculations. Option D is an example of empirical probability because it is based on past observations of employee retention.

Quiz 4: Introduction to Probability

Which of the following sets of outcomes described below in I and II represent mutually exclusive events? I. "Your final course grade is an A"; "Your final course grade is a B." II) "Your final course grade is an A"; "Your final course grade is a Pass."

The intersection of events A = {apple pie, peach pie, pumpkin pie} and B = {cherry pie, blueberry pie, pumpkin pie} is ________.

The intersection of events A and B, denoted by A ∩ B, ________.

An experiment consists of tossing three fair coins. What is the probability of tossing two tails?

Which of the following sets of outcomes described below in I and II represent mutually exclusive events? I. "You are a freshman"; "You are a senior." II) "You are a freshman"; "You are an underclassman."

Mutually exclusive and collectively exhaustive events ________.

The odds for encountering rain on a 500-mile car trip are three to one. What is the probability of rain on this trip?

An analyst has a limit order outstanding on a stock. He argues that the probability that the order will execute before the close of trading is 0.20. Thus, the odds for the order executing before the close of trading are

The complement of an event A, within the sample space S, is the event consisting of ________.

An analyst believes the probability that U.S. stock returns exceed long-term corporate bond returns over a five-year period is based on personal assessment. This type of probability is best characterized as a(n) ________.

A probability based on logical analysis rather than on observation or personal judgment is best referred to as a(n) ________.

The union of events A and B, denoted by A ∪ B, ________.

After extensive research, an analyst asserts that there is an 80% chance that ABC Corporation will beat its EPS forecast. Analogously, the odds for the company beating its EPS forecast are ________.

For the sample space S = {apple pie, cherry pie, peach pie, pumpkin pie}, what is the complement of A = {pumpkin pie, cherry pie}?

Let P(A) = 0.6, P(B) = 0.3, and P(A ∪ B^C) = 0.1. Calculate P(A ∩B) .

The odds against winning $1.00 in the lottery are 19 to 1. What is the probability of winning $1.00 in the lottery?

Which of the following represents a subjective probability?

The union of events A = {apple pie, peach pie, pumpkin pie} and B = {cherry pie, blueberry pie, pumpkin pie} is ________.

Assume the sample space S = {win, loss}. Select the choice that fulfills the requirements of the definition of probability.

Which of the following represents an empirical probability?