Question

Olympic athletes have to undergo anti-doping tests to verify that they did not use performance enhancing drugs during the competition.

• According to page 43 of the World Anti Doping Agency (WADA) 2017 Anti-doping Testing figures 0.9% of testing conducted on Olympic athletes in the "Athletics" sport, which includes sprinting, yields an AAF - Adverse Analytical Finding - i.e. a positive result.
• It is harder to find information on the reliability of the tests conducted, so for the sake of this exercise, we will assume that the tests in question have a false negative rate of 5% and a false positive rate of 0.5%.

Use this information to calculate the probability that an Olympic sprinter who tests positive has actually used performance enhancing drugs. For this exercise, define the following events:

• T⁺: getting an AAF on a test (i.e. testing positive)
• T^-: not getting an AAF on a test (i.e. testing negative)
• U: using a performance enhancing drug (i.e. being a user)
• C: not using a performance enhancing drug (i.e. being clean)

Answer 1

P(T+) = 0.9%

P(T-) = 1 - 0.9% = 99.1%

The truth probability table for this scenario is shown below:

	Tests +ve	Tests -ve
Used Drugs	P(TP)=0.4%	P(FN) = 5%
Didn't use	P(FP) = 0.5%	P(TN)=94.1%
Total	P(T+): 0.90%	P(T-): 99.1%

where P(True +ve) is calculated as: P(T+) - P(FP), and True -ve as P(T-) - P(FN)

We need to find P(Used drugs / Tests positive) = P(Used drugs & Test Positive) / P(Test Positive)

= P(TP) / P(T+)

= 0.4%/0.9% = 0.4444 = 44.44%