Question

Suppose 5% of athletes at the Olympic games dope, i.e., use performance enhancing drugs. The International Olympic Committee has a test that correctly identifies athletes who have doped 90% of the times. This test also correctly identifies athletes who have not doped 96% of the times. The test came out positive for an athlete. What is the chance that he/she has not doped?

Answer 1

P(using dope)= 0.05

P(test positive | using dope)= 0.90

P(test negative | not using dope)= 0.96

Thus,

P(test positive | no using dope) = 1- P(test positive | using dope)

= 1- 0.90

= 0.10

P(not using dope)= 1-0.05

= 0.95

P(using dope | test positive)= P(test positive | using dope) * P(using dope) / P(test positive)

= P(test positive | using dope) * P(using dope) / [P(test positive | using dope) * P(using dope) + P(test positive | not using dope) * P(not using dope)]

= 0.90*0.05 / [0.90*0.05 + 0.10*0.95]

= 0.045 / (0.045+0.095)

= 0.045 / 0.14

= 0.32142857142 or 32.14%