False Negative/False Positive

Question

A drug testing company advertises a test for steroids that is 93.5% effective at detecting steroid use. It has a true negative effectiveness of 90.5%. Suppose 6% of 10,000 professional cyclists are believed to be using steroids.

Suppose a cyclist tests negative for steroid use. What is the probability that he's using steroids?

Answer 1

If a test is 93.5% effective at detecting steroid use, that means that 6.5% of those who are using steroids will test negative. If the test has a 90.5% true negative effectiveness, that means that 9.5% of those who are not using will test positive. (That would be an unacceptably high false positive rate in my opinion, but that's for another problem.) Under the assumption that 6% of cyclists are actually using steroids, that means that 6.5% of 6% = 0.39% of the cyclist population will have false negative tests. If 94% of the cyclists are not using, then 90.5% of 94% = 85.07% of the cyclists will have true negative tests. Since our randomly selected cyclist has a negative test, he must be in one of these two populations. Therefore, the probability that he is indeed using steroids is 39 / (8507 + 39) = 0.456%.