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

1. 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.
2. 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)).
3. 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.
4. 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.
5. 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.

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 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.