Question

Consider a test for detecting if an athlete has used a performance-enhancing drug. Let θ be the likelihood that the drug test gives the correct result. That is, the probability of drug test being positive (R = 1) when they actually did (D = 1), is θ. And the probability it reports no drug use when indeed there was none is also θ. More formally, P(R=1|D=1) = P(R=0|D=0) = θ Suppose also that the proportion of athletes in a particular sport that use the drug is α, where 0 ≤ α ≤ 1.

(a) Suppose an athlete tests positive. Derive the posterior probability that the athlete used the drug, and simplify it in terms of θ and α.

(b) Suppose θ = .99. Suppose 1000 athletes are tested, and they’re all ’clean’. What is the expected number of false positives?

Answer 1

It is given that . Also

So, using complementary probability,

Using total probability theorem,

a) The posterior probability that the athlete used the drug given the athlete tests positive is the conditional probability,

b) False positives (testing positive given the athlet is clean) is the conditional probability . Thus the probability of false positive in a test is . The number of athlets out of who tests false positive has binomial distribution.

The PMF of is  . The expected value of the above Binomial distribution is