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Bayes' theorem is a formula that allows us to compute conditional probabilities. In this context, it can be stated as: P(A | B) = P(B | A) * P(A) / P(B) where A represents an event or a condition and B represents another event or condition related to A. For our exercise, A will represent being a baby boomer (or a pre-boomer) and B will represent answering in the affirmative.

To apply Bayes' theorem, we first need to compute the probabilities for each event: - P(Baby Boomer) or P(A): There are 400 baby boomers out of a total of 1000 respondents (400+600), so P(A) = 400/1000 = 0.4. - P(Pre-Boomer) or P(A'): There are 600 pre-boomers out of a total of 1000 respondents, so P(A') = 600/1000 = 0.6. - P(Affirmative | Baby Boomer) or P(B|A): Out of the 400 baby boomers, 212 answered in the affirmative, so P(B|A) = 212/400 = 0.53. - P(Affirmative | Pre-Boomer) or P(B|A'): Out of the 600 pre-boomers, 198 answered in the affirmative, so P(B|A') = 198/600 = 0.33.

Next, we need to compute the probability that a respondent answered in the affirmative, P(B). We can do this using the law of total probability: P(B) = P(B | A) * P(A) + P(B | A') * P(A') P(B) = (0.53 * 0.4) + (0.33 * 0.6) = 0.212 + 0.198 = 0.410

Now, we can apply Bayes' theorem to find the probability that a respondent chosen at random from those surveyed answered the question in the affirmative is a baby boomer and the probability that they are a pre-boomer: P(Baby Boomer | Affirmative) = P(Affirmative | Baby Boomer) * P(Baby Boomer) / P(Affirmative) = (0.53 * 0.4) / 0.410 ≈ 0.517 P(Pre-Boomer | Affirmative) = P(Affirmative | Pre-Boomer) * P(Pre-Boomer) / P(Affirmative) = (0.33 * 0.6) / 0.410 ≈ 0.483 So, if a respondent chosen at random from those surveyed answered the question in the affirmative, there is a probability of approximately 51.7% that he or she is a baby boomer, and a probability of approximately 48.3% that he or she is a pre-boomer.