A life insurance salesman finds that, of all the sales he...

A life insurance salesman finds that, of all the sales he makes, $70 \%$ are to people who already own policies. He also finds that, of all contacts for which no sale is made, $50 \%$ already own life insurance policies. Furthermore, $40 \%$ of all contacts result in sales. What is the probability that a sale will be made to a contact who already owns a policy?

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Key Concepts

Conditional Probability

This concept refers to the probability of an event occurring given that another event has already occurred. It is central to problems where certain conditions or events affect the likelihood of subsequent events, such as determining the probability of making a sale to someone who already owns a policy given the overall sale rates and ownership percentages.

Joint Probability

Joint probability involves calculating the probability of two events happening at the same time. In the context of the question, it is used to find the probability that a contact both results in a sale and already owns a policy, by combining the probability of making a sale with the conditional probability of owning a policy within that sale subset.

Law of Total Probability

This principle allows one to break down the probability of an event into several disjoint cases that cover all possible scenarios. It is relevant here as the overall probability for events such as sales or policy ownership can be partitioned into cases where a sale is made and where a sale is not made, helping in the computation of combined probabilities across these scenarios.

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