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Chapter 7: Problem 20

In a survey of 2000 adults \(50 \mathrm{yr}\) and older of whom \(60 \%\) were retired and \(40 \%\) were preretired, the following question was asked: Do you expect your income needs to vary from year to year in retirement? Of those who were retired, \(33 \%\) answered no, and \(67 \%\) answered yes. Of those who were pre-retired, \(28 \%\) answered no, and \(72 \%\) answered yes. If a respondent in the survey was selected at random and had answered yes to the question, what is the probability that he or she was retired?

Short Answer

Expert verified
Given a person in the survey answered 'yes' to the question, there is a \(58.2\%\) chance that they are retired.

Step by step solution

01

Understand and define the problem

We are given a number of probabilities and asked to find a conditional probability. The question is, given that someone answered "yes", what is the probability they are retired? In other words, we want to find \( P(\text{{Retired}} | \text{{Yes}}) \).
02

Identify the Given Probabilities

From the problem, we have: \ - The Prior probability, \(P(\text{{Retired}})\) = 0.60 and \(P(\text{{Pre-retired}})\) = 0.40 - The Likelihood, \(P(\text{{Yes}} | \text{{Retired}})\) = 0.67 and \(P(\text{{Yes}} | \text{{Pre-retired}})\) = 0.72
03

Calculate the Marginal Likelihood

This is the probability that a randomly chosen person said "yes", which is given by: \(P(\text{{Yes}}) = P(\text{{Yes}}|\text{{Retired}}) \cdot P(\text{{Retired}}) + P(\text{{Yes}}|\text{{Pre-retired}}) \cdot P(\text{{Pre-retired}})\). Plugging the numbers in, we find \( P(\text{{Yes}}) = 0.67 \cdot 0.60 + 0.72 \cdot 0.40 = 0.402 + 0.288 = 0.69 \).
04

Apply Bayes Theorem

Finally, we can find the required probability as follows: \(P(\text{{Retired}} | \text{{Yes}}) = \frac{P(\text{{Yes}} | \text{{Retired}}) \cdot P(\text{{Retired}})}{P(\text{{Yes}})}\). We can substitute in our calculated probabilities to find that \(P(\text{{Retired}} | \text{{Yes}}) = \frac{0.67 \cdot 0.60}{0.69} = 0.582 ~\text{or} ~58.2 \%\) So, given a person in the survey answered 'yes' to the question, there is a 58.2% chance that they are retired.

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Most popular questions from this chapter

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