Ch. 3 - Probability

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 3 - Probability

Problem 3.2.8b

Chapter 3, Problem 3.2.8b

Finding Conditional Probabilities In Exercises 7 and 8, use the table to find each conditional probability.
8. Retirement Savings The table shows the results of a survey in which 250 male and 250 female workers ages 25 to 64 were asked if they contribute to a retirement savings plan at
work.
b. Find the probability that a randomly selected worker is female, given that the worker contributes to a retirement savings plan at work.

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the conditional probability that a randomly selected worker is female, given that the worker contributes to a retirement savings plan at work. Conditional probability is calculated using the formula P(A|B) = P(A ∩ B) / P(B), where P(A|B) is the probability of A given B.

Step 2: Identify the relevant data from the table. The total number of workers who contribute to a retirement savings plan is 259 (this is P(B)). Out of these, the number of female workers who contribute is 143 (this is P(A ∩ B)).

Step 3: Write the formula for conditional probability. Using the formula P(A|B) = P(A ∩ B) / P(B), substitute the values: P(Female | Contribute) = Number of females who contribute / Total number of workers who contribute.

Step 4: Substitute the values into the formula. From the table, the number of females who contribute is 143, and the total number of workers who contribute is 259. So, P(Female | Contribute) = 143 / 259.

Step 5: Simplify the fraction if needed. The result will give the conditional probability that a randomly selected worker is female, given that they contribute to a retirement savings plan.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conditional Probability

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. It is denoted as P(A|B), which reads as the probability of event A occurring given that event B has occurred. This concept is crucial for understanding how probabilities can change based on additional information, such as knowing whether a worker contributes to a retirement savings plan.

Joint Probability

Joint probability is the probability of two events happening at the same time. In the context of the provided table, it refers to the probability of a worker being both female and contributing to a retirement savings plan. This is calculated by dividing the number of females who contribute by the total number of surveyed workers, providing a foundational understanding for calculating conditional probabilities.

Marginal Probability

Marginal probability is the probability of a single event occurring without consideration of other events. In this case, it refers to the overall probability of selecting a female worker from the total surveyed population. It is calculated by dividing the total number of females by the total number of workers, and it serves as a baseline for understanding how conditional probabilities are derived from the overall distribution of the data.

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Related Practice

Textbook Question

25. Working from Home The table shows the results of a survey that asked 1811 people how often they work from home. A person is selected at random from the sample. Find the probability of each event.

b. The person is female or does not work from home.

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Textbook Question

"Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.

23. Celebrities as Role Models In a sample of 1103 probable voters, three out of four say they would like entertainers to address social and political issues. Two probable voters are selected at random. (Source: The Hollywood Reporter)

b. Find the probability that neither probable voter would like entertainers to address social and political issues."

Textbook Question

Politics The responses of 1500 U.S. adults to a survey that asked them to state their own political viewpoints are shown in the Pareto chart. Find the probability of each event.(Adapted from YouGov)

b. Randomly selecting a person from the sample who is conservative or very conservative

Textbook Question

Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.

28. Blood Types The probability that a Latinx American person in the United States has type A+ blood is 29%. Four Latinx American people in the United States are selected at random. (Source: American National Red Cross)

b. Find the probability that none of the four have type A+ blood.

Textbook Question

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b. What is the probability that none of the offices are filled by members of the debate team?

Textbook Question

22. Brexit A survey asked 1115 British adults how Britain's decision to leave the European Union has impacted the country. The results are shown in the Pareto chart. Find the

probability of each event. (Adapted from Ipsos)

b. Randomly selecting a British adult who feels that the move has had a very negative impact on Great Britain