2014 AIME II Problems/Problem 2
Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of men. For each of the three factors, the probability that a randomly selected man in the population has only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probability that a randomly selected man has exactly these two risk factors (but not the third) is 0.14. The probability that a randomly selected man has all three risk factors, given that he has A and B is . The probability that a man has none of the three risk factors given that he does not have risk factor A is , where and are relatively prime positive integers. Find .
We first assume a population of to facilitate solving. Then we simply organize the statistics given into a Venn diagram.
Let be the number of men with all three risk factors. Since "the probability that a randomly selected man has all three risk factors, given that he has A and B is ," we can tell that , since there are people with all three factors and 14 with only A and B. Thus .
Let be the number of men with no risk factors. It now follows that The number of men with risk factor A is (10 with only A, 28 with A and one of the others, and 7 with all three). Thus the number of men without risk factor is 55, so the desired conditional probability is . So the answer is .
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