Drug tests are a common way to detect doping among athletes. Lets consider a pro

Question

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.

i. Suppose someone has a positive test, what is the probability that they are doping? [HINT: Denote testing positive by P and doping by D. What are the probabilities given in the text above? You may find using a Venn Diagram helps make this easier, but the key is to think of testing positive and doping as two separate events.]

ii. Suppose someone has a positive test, what is the probability that they are not doping?

iii. 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. 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 Pr(P | D’)?

Explanation / Answer

Ans:

P(P/D)=0.95

P(D)=0.03

P(D')=1-0.03=0.97

P(P)=0.04

P(P')=1-0.04=0.96

P(P/D')=0.0115/0.97=0.0119

i)P(D/P)=P(P/D)*P(D)/[P(P/D)*P(D)/+P(P/D')*P(D')]

=0.95*0.03/[0.95*0.03+0.0119*0.97]

=0.0285/[0.0285+0.0115]

=0.7125

ii)P(D'/+ve)=1-0.7125=0.2875

iii)P(P/D')=0.0115/0.97=0.0119

Drug not drug marginal positive 0.0285 0.0115 0.04 negative 0.0015 0.9585 0.96 marginal 0.03 0.97 1

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