"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.

a. Find the probability that a randomly selected worker contributes to a retirement savings plan at work, given that the worker is male.
"

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the conditional probability that a randomly selected worker contributes to a retirement savings plan at work, given that the worker is male. Conditional probability is calculated using the formula: P(A|B) = P(A ∩ B) / P(B).
2. Step 2: Identify the relevant data from the table. From the table, we see that there are 116 males who contribute to a retirement savings plan and a total of 250 males.
3. Step 3: Define the events. Let A be the event 'worker contributes to a retirement savings plan' and B be the event 'worker is male'. The intersection of A and B (A ∩ B) corresponds to the number of males who contribute, which is 116.
4. Step 4: Calculate P(B). The probability of selecting a male worker, P(B), is the total number of males divided by the total number of workers: P(B) = 250 / 500.
5. Step 5: Calculate P(A|B). Using the formula for conditional probability, P(A|B) = P(A ∩ B) / P(B). Substitute the values: P(A ∩ B) = 116 / 500 and P(B) = 250 / 500. Simplify the expression to find the conditional probability.

Key Concepts

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 is true. In this context, we are interested in the probability that a worker contributes to a retirement savings plan, given that the worker is male.

Joint Probability

Joint probability is the probability of two events happening at the same time. It is calculated by multiplying the probabilities of the individual events if they are independent. In the table, the joint probability of being male and contributing to a retirement savings plan can be found by looking at the number of males who contribute relative to the total number of surveyed workers.

Marginal Probability

Marginal probability is the probability of an event occurring without any conditions on other events. It is derived from the total counts in a probability distribution. In this case, the marginal probability of a worker being male is calculated by dividing the total number of males by the overall total number of workers surveyed, which helps in understanding the context of the conditional probability.