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Suppose that 2% of all Olympic athletes use steroids or other performance-enhancing drugs. Because drug

use is a serious violation of Olympic rules, the International Olympic Committee has implemented random

drug tests. Let us assume that the test will report positive with probability 0.95 if the athlete uses drugs,

and with probability 0.2 if the athlete does not use drugs.

a) A randomly selected athlete has a positive drug test. What is the probability that he uses drugs?

b) A randomly selected athlete has a negative drug test. What is the probability that she uses drugs?

c) In fact, an athlete is typically asked to give two samples, A and B, which are tested independently.

Assuming that the results of the A and B tests truly are independent, and that both of an athlete’s samples

are positive, what is the probability that he uses drugs?

d) Given that an athlete’s A test is positive, what is the probability that her B test will also be posi-

tive?

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# Bayes Theorem

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