Drug tests are a common way to detect doping among athletes. Lets consider a professional cycling drug test in detail now. Suppose we know the probability of getting a positive outcome if someone uses drugs is 95%. We also know based on an anonymous survey that 3% of cyclists are using drugs or doping. (Take 3% as the true probability that a cyclist dopes). The cycling federation performs a surprise drug test on all racers and finds that 4% of outcomes are positive.

(a) (1 point) Suppose someone has a positive test, what is the probability that they are doping? (If you are stuck see the hint below for how to begin.)

(b) (1 point) Suppose someone has a positive test, what is the probability that they are not doping?

(c) (2 points) Obviously, this probability is too high to penalize all sportsmen with a positive outcome in it. This is one reason why you often have to fail more than one test before you are sanctioned and why the sports agencies generally do not publicly announce positive results unless you have crossed their threshold that they set on the number of positive tests you need to have.

(d) (2 points) Its important to note, that this last question is not the same thing as the false positive rate. What is the probability of a false positive test in this case? i.e., what is P r(P|D’ )?

(e) (2 points) The false positive rate is reassuring. If you are a cyclist who does not dope, you can be pretty sure that you will not face sanctions at your next race, particularly if it requires two positive tests to get a sanction. However, just like in our first class, the probability of you being sanctioned may be incredibly low, but the probability that at least one of the 200 non-doping riders being tested will be sanctioned is nontrivial (2.84%).