Question

All athletes at the Olympic games are tested for performance-enhancing steroid drug use. The imperfect test gives positive results (indicating drug use) for 90% of all steroid-users but also for 2% of those who do not use steroids (that is, the false-positive rate is 2%). Suppose that 4% of all registered athletes use steroids. If an athlete is tested negative, what is the probability that the test is wrong and he or she actually uses steroids?

Answer 1

0.4%

Explanation:

We have two independent events

1. The athlete who is being tested for drugs actually is using steroids.
2. The test went wrong indicating a false result.

The probability of 1) to happen is 0.04 because "4% of all registered athletes use steroids".

The probability of 2) ti happen is 0.1 because "The imperfect test gives positive results (indicating drug use) for 90% of all steroid-users"

Then the probability of 1) and 2) to happen is given by

`P=P(1)*P(2)=(0.04)(0.1)=0.004`=0.4%