<p>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 <img alt="$\frac{1}{3}$" height="37" src="https://latex.artofproblemsolving.com/4/e/1/4e1965f8d70832ded66434c75a7bd52b75090af0.png" width="11" />. The probability that a man has none of the three risk factors given that he does not have risk factor A is <img alt="$\frac{p}{q}$" height="36" src="https://latex.artofproblemsolving.com/7/3/4/7347a1d5586e82c832f61ea799d89945ed2837fc.png" width="11" />, where <img alt="$p$" height="11" src="https://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png" width="10" /> and <img alt="$q$" height="11" src="https://latex.artofproblemsolving.com/0/6/1/0615acc3725de21025457e7d6f7694dab8e2f758.png" width="8" /> are relatively prime positive integers. Find <img alt="$p+q$" height="14" src="https://latex.artofproblemsolving.com/1/7/0/170200756543596efeb46fca1cf29a617e8c9192.png" width="41" />.</p>

Question No: 383

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 $\frac{1}{3}$ . The probability that a man has none of the three risk factors given that he does not have risk factor A is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .

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Tags: Probability

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