Question

In a survey to determine the opinions of Americans on health insurers, 400 baby boomers and 600 pre-boomers were asked this question: Do you believe that insurers are very responsible for high health costs? Of the baby boomers, 232 answered in the affirmative, whereas 192 of the pre-boomers answered in the affirmative. If a respondent chosen at random from those surveyed answered the question in the affirmative, what is the probability that he or she is a baby boomer? A pre-boomer? (Round your answers to two decimal places.)

baby boomer ?
pre-boomer ?

Answer 1

Let events
B=baby boomer
~B=pre-boomer (complement of B)
A=answered in affirmative

then by the law of total probability:
P(A)=P(A∩B)+P(A∩~B)
=P(A|B)*P(B)+P(A|~B)*P(~B)
=232/400*(400/1000)+192/600*(600/1000)
=0.424

using Bayes theorem,
P(B|A)=P(A|B)P(B)/P(A)
=(232/400)*0.4/(0.424)
=0.5472
or
P(boomer|affirmative answer) = 0.5472

I'll leave it to you to find
P(pre-boomer|affirmative answer) by finding the complement.