Question: An actuary is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of women. For each of the three factors, the probability is 0.1 that a woman in the population only has this risk factor (and no others). For any two of three factors, the probability is 0.12 that she has exactly two of these risk factors (but not the other). The probability that a woman has all three risk factors given that she has A and B, is (1/3). What is the probability that a woman has none of the three risk factors, given that she does not have risk factor A?
My attempt: I wrote the "probability that a woman has none of the three risk factors, given that she does not have risk factor A" as Pr(A'andB'andC'|A') as Pr(A'andB'andC'andA')/Pr(A') which just simplifies to Pr(A'andB'andC')/Pr(A') where Pr(A') = (1-.1) = .9. I'm not entirely sure where to go on from there. I also tried to draw a Venn Diagram with three intersecting circles where Pr(AandB'andC') = .1 (same for B and C), but that didn't really get me anywhere The answer is 0.467 (rounded). Can you guys please show me what I'm doing wrong or what I should be doing?
Thank you guys so much!