A record-store owner assesses customers entering the store...

A record-store owner assesses customers entering the store as high school age, college age, or older, and finds that of all customers $30 \%, 50 \%$, and $20 \%$, respectively, fall into these categories. The owner also found that purchases were made by $20 \%$ of high school age customers, by $60 \%$ of college age customers, and by $80 \%$ of older customers. a. What is the probability that a randomly chosen customer entering the store will make a purchase? b. If a randomly chosen customer makes a purchase, what is the probability that this customer is high school age?

A record-store owner assesses customers entering the store as high school age, college age, or older, and finds that of all customers $30 \%, 50 \%$, and $20 \%$, respectively, fall into these categories. The owner also found that purchases were made by $20 \%$ of high school age customers, by $60 \%$ of college age customers, and by $80 \%$ of older customers.
a. What is the probability that a randomly chosen customer entering the store will make a purchase?
b. If a randomly chosen customer makes a purchase, what is the probability that this customer is high school age?

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Key Concepts

Bayes' Theorem

Bayes' Theorem is a method for updating the probability estimate of an event based on new information. In scenarios like the given problem, it is used to reverse conditional probabilities; that is, it allows the calculation of the probability of a customer being in a specific age group given that the customer made a purchase.

Law of Total Probability

This principle allows one to compute the overall probability of an event by breaking it down into several mutually exclusive cases or categories. In the context of the problem, the overall purchase probability is calculated by considering the contribution of each age group and then summing the weighted probabilities based on the fraction of customers in each category.

Conditional Probability

Conditional probability measures the likelihood of an event occurring given that another event has already occurred. For instance, the problem asks for the probability that a purchaser is a high school age customer, which is a conditional probability calculated using the known probabilities of being a high school age customer and of making a purchase.

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