Two doping tests both come back positive

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Only 1\% of athletes in a sport actually dope. A drug test detects 95\% of true dopers (sensitivity) and correctly clears 90\% of clean athletes, so it false-alarms on 10\% of the clean. An athlete is tested twice, with the tests independent given the truth. Both are positive. What is the probability the athlete is doping?

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Imagine 10{,}000 athletes: 100 dope, 9{,}900 are clean. Multiply the independent per-test rates. - Doping, both positive: 100 	imes 0.95 	imes 0.95 = 90.25. - Clean, both positive: 9{,}900 	imes 0.10 	imes 0.10 = 99. Of the 90.25 + 99 = 189.25 double-positives, 90.25 are truly doping: P(	ext{doping} \mid ++) = \frac{90.25}{189.25} \approx 0.477 = 48\%. A single positive here is very weak (under 9\%) because doping is so rare, but the second independent positive nearly reaches a coin flip. Takeaway. With an extremely low base rate, even two positives may not clear 50\%; a lower false-posi…

Two doping tests both come back positive

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