Ch 4 Hw

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

0 An addition rule is P(A∪B) = P(A) + P(B) − P(A∩B). A complement rule is P(A) = 1 − P(Ac). If P(A∪B)C = 0.10 the P(A∪B) = 0.90 P(A∪B) = P(A) + P(B) − P(A∩B) So, 0.90 = 0.60 + 0.30 − P(A∩B) and P(A∩B) = 0

Peter applied to an accounting firm and a consulting firm. He knows that 30% of similarly qualified applicants receive job offers from the accounting firm, while only 20% of similarly qualified applicants receive job offers from the consulting firm. Assume that receiving an offer from one firm is independent of receiving an offer from the other. What is the probability that both firms offer Peter a job?

0.06 The joint probability for two independent events is the product of the individual probabilities. P(Accounting∩Consulting) = 0.30 × 0.20 = 0.06.

The contingency table below provides frequencies for the preferred type of exercise for people under the age of 35, and those 35 years of age or older. Find the probability that an individual prefers biking given that he or she is 35 years old or older. Age Group | Preferred Form of Exercise | Total | Running Biking Swimming | Under 35 Years | 157 | 121 | 79 | 375 35 Years or Older | 45 | 27 | 87 | 159 Total | 202 | 148 | 166 | 516

0.1698 The contingency table shows frequencies for two qualitative or categorical variables, x and y, where each cell represents a mutually exclusive combination of the pair of x and y values. A more convenient way of calculating relevant probabilities is to convert the contingency table to a joint probability table. P(Biking∣∣35 years or older)=27/159=0.1698

Alison has all her money invested in two mutual funds, A and B. She knows that there is a 30% chance that fund A will rise in price, and a 70% chance that fund B will rise in price given that fund A rises in price. What is the probability that both fund A and fund B will rise in price?

0.21 The conditional probability is calculated as P(B∣∣A)=P(A∩B)P(A).P(B|A)=P(A∩B)/P(A). Given: P(A) = 0.30 and P(B|A) = 0.70 P(A∩B) = P(A|B) × P(B) = P(B|A) × (A) = 0.70 × 0.30 = 0.21.

The contingency table below provides frequencies for the preferred type of exercise for people under the age of 35 and those 35 years of age or older. Find the probability that an individual prefers running and is under 35 years of age. Age Group | Preferred Form of Exercise | Total | Running Biking Swimming | Under 35 Years | 157 | 121 | 79 | 375 35 Years or Older | 45 | 27 | 87 | 159 Total | 202 | 148 | 166 | 516

0.3042 The contingency table shows frequencies for two qualitative or categorical variables, x and y, where each cell represents a mutually exclusive combination of the pair of x and y values. A more convenient way of calculating relevant probabilities is to convert the contingency table to a joint probability table.

Let P(A) = 0.4, P(B|A) = 0.5, and P(B|Ac) = 0.25. Compute P(B).

0.35 The total probability rule conditional on two events B and Bc is defined as P(A)=P(A∩B)+P(A∩Bc)P(A)=P(A∩B)+P(A∩Bc) P(A|B)P(B)+P(A|Bc)P(Bc)P(A|B)P(B)+P(A|Bc)P(Bc) P(B) = 0.40 × 0.50 + (1 − 0.40) × 0.25 = 0.35

A company is bidding on two projects, A and B. The probability that the company wins project A is 0.40 and the probability that the company wins project B is 0.25. Winning project A and winning project B are independent events. What is the probability that the company does not win either project?

0.45 The addition rule is calculated as P(A∪B)=P(A)+P(B)−P(A∩B).P(A∪B)=P(A)+P(B)−P(A∩B). A complement rule is P(A)=1−P(Ac).P(A)=1−P(Ac). P(A∪B) = 0.40 + 0.25 − 0.1 = 0.55 Neither = 1 − P(A∪B) = 1 − 0.55 = 0.45

Let P(A∩B)=0.3P(A∩B)=0.3 , and P(A∩Bc)=0.15P(A∩Bc)=0.15 . Compute P(A).

0.45 The total probability rule conditional on two events B and Bc is defined as P(A)= P(A∩B)+P(A∩Bc) = P(A)= P(A∩B) + P(A∩Bc) =P(A|B) P(B)+ P(A|Bc) P(Bc) P(A) = 0.30 + 0.15 = 0.45

A company is bidding on two projects, A and B. The probability that the company wins project A is 0.40 and the probability that the company wins project B is 0.25. Winning project A and winning project B are independent events. What is the probability that the company wins project A or project B?

0.55 The addition rule is calculated as P(A∪B)=P(A)+P(B)−P(A∩B)P(A∪B)=P(A)+P(B)−P(A∩B) . Since the events are independent, then the P(A&B) = P(A) × P(B) = 0.4 × 0.25 = 0.1 Hence P(A or B) = 0.4 + 0.25 − 0.1 = 0.55

Alison has all her money invested in two mutual funds, A and B. She knows that there is a 40% chance that fund A will rise in price, and a 60% chance that fund B will rise in price given that fund A rises in price. There is also a 20% chance that fund B will rise in price. What is the probability that neither fund will rise in price?

0.64 The addition rule is calculated as P(A∪B)=P(A)+P(B)−P(A∩B)P(A∪B)=P(A)+P(B)−P(A∩B) . By the complement rule, P(A)=1−P(Ac)P(A)=1−P(Ac) . Given: P(A) = 0.40 and P(B|A) = 0.60, P(B) = 0.20 P(neither) = 1 − P(either) = 1 − (A∪B) = 1 − (0.40 + 0.20 − 0.40 × 0.60) = 1 − 0.36 = 0.64

The following probability table shows probabilities concerning Favorite Subject and Gender. What is the probability of selecting an individual who is a female or prefers science? Gender | Favorite Subject | Total | Math English Science | Male | 0.2 | 0.05 | 0.175 | 0.425 Female | 0.1 | 0.325 | 0.150 | 0.575 Total | 0.3 | 0.375 | 0.325 | 1.0

0.750 The contingency table shows frequencies for two qualitative or categorical variables, x and y, where each cell represents a mutually exclusive combination of the pair of x and y values. A more convenient way of calculating relevant probabilities is to convert the contingency table to a joint probability table. P(Female∪Science) = 0.575 + 0.325 − 0.150 = 0.750

There is a 25% chance that a customer who purchases milk will also purchase bread. The probability of a milk purchase is 70% and the probability of a bread purchase is 50%. What is the probability that a customer will purchase bread given that he/she buys milk?

36% P(milk and bread) = 0.25; P(milk) = 0.7; P(bread) = 0.5. P(bread | milk) = P(milk and bread)/P(milk) = 0.25/0.7 = 36%

The likelihood of Company A's stock price rising is 20%, and the likelihood of Company B's stock price rising is 30%. Assume that the returns of Company A and Company B stock are independent of each other. The probability that the stock price of at least one of the companies will rise is ______.

44% The addition rule is calculated as P(AUB)=P(A)+P(B)−P(A∩B)P(AUB)=P(A)+P(B)−P(A∩B) . P(A∪B) = 0.20 + 0.30 − 0.20 × 0.30 = 0.44.

Which of the following represents a subjective probability?

A skier believes she has a 10% chance of winning a gold medal. For well-defined problems an a priori probability can be calculated by reasoning about the problem. A subjective probability is based on personal experience and judgment.

Given an experiment in which a fair coin is tossed three times, the sample space is S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Event A is defined as tossing one head (H). What is the event Ac and what is the probability of this event?

Ac = {TTT, HHT, HTH, THH, HHH}; P(Ac) = 0.625 The complement of event A, Ac, is the event consisting of all outcomes in the sample space S that are not in A. Of the 8 outcomes, we only want those that have two heads or no heads. This makes 5 outcomes and 5/8 = 0.625.

Which of the following represents an empirical probability?

Based on past observation, a manager believes there is a three-in-five chance of retaining an employee for at least one year. We use the relative frequency to calculate the empirical probability of event A as P(A)= the number of outcomes in A/the number of outcomes in S ⋅

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

Only I represents mutually exclusive events. Events are mutually exclusive if they do not share any common outcome of a random experiment.

If A and B are independent events, which of the following is correct?

P(AB)=P(A) For two independent events A and B: P(A|B)=P(A)P(A|B)=P(A) . P(A∩B) = 0 and P(A∪B) = P(A) + P(B) both apply to mutually exclusive events.

When some objects are randomly selected, which of the following is true?

The order in which objects are selected does not matter in combinations. The order in which x objects are arranged does not matter for combinations.

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

contains outcomes that are both in A and B The intersection of two events, A∩BA∩B , is the event consisting of all outcomes in both A and B.

Events are collectively exhaustive if _____________.

they contain all outcomes of an experiment. Events are collectively exhaustive if all possible outcomes of a random experiment are included in the events.

Find the missing values marked xx and yy in the following contingency table. Age Group | Preferred Form of Exercise | Total | Running Biking Swimming | Under 35 Years | 157 | 121 | yy | 375 35 Years or Older | 45 | xx | 87 | 159 Total | 202 | 148 | 166 | 516

xx = 27, yy = 79 The contingency table shows frequencies for two qualitative or categorical variables, x and y, where each cell represents a mutually exclusive combination of the pair of x and y values. A more convenient way of calculating relevant probabilities is to convert the contingency table to a joint probability table.

In an experiment in which a coin is tossed twice, which of the following represents mutually exclusive and collectively exhaustive events?

{TT, HH} and {TH, HT} Events are mutually exclusive if they do not share any common outcome of a random experiment. Events are collectively exhaustive if all possible outcomes of a random experiment are included in the events.

Which of the following are mutually exclusive events of an experiment in which two coins are tossed?

{TT, HH} and {TH} Events are mutually exclusive if they do not share any common outcome of a random experiment

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

{apple pie, peach pie} The complement of event A, Ac, is the event consisting of all outcomes in the sample space S that are not in A.