Drug Testing
Solution part 1: false
positive: 99%*1%=0.99%, false negative:
1%*1%=0.01%.)
Solution part 2: true
positive: 1%*99%=0.99%, false positive: = 99%*1% = 0.99%, positive
= true positive + false positive = 1.98%, P(drug | +) = 0.99%/1.98%
= 50%.)
Solution part 3: If both
athletes dopes, then each has 1%*(100%-1%*50%)=0.95% chance of
winning if s/he luckily passes the test, unless the other athlete
also passes and beats him/her. If neither dopes, then each has
99%*(99%*50% + 1%)= 49.995% chance of winning by passing the drug
test and beating the opponent, or if the opponent fails the drug
test. If one athlete dopes and the other doesn't, then the doped
athlete has 1%*(99%*75% + 1%) = 0.7525% chance of winning by
passing the drug test and beating the opponent, or if the opponent
fails the drug test. Similarly, the undoped athlete has
99%*(1%*25%+99%) = 98.2575% chance of winning. So the payoff
matrix is now
| A/B (%) |
Drug |
No Drug |
| Drug |
0.95/0.95 |
0.7525/98.2575
|
| No Drug |
98.2575/0.7525
|
49.995/49.995
|
Solution part 4: If
both athletes dope, then A has 1%*50% = 0.5% chance of winning.
Since B cannot be disqualified, s/he will have 1-0.5%=99.5% chance
of winning. If only A dopes, then s/he has 1%*75%=0.75% chance of
winning. If only B dopes, then A has 99%*25% = 24.75% chance of
winning. If neither dopes, then A has 99%*50% chance of
winning.
| A/B (%) |
Drug |
No Drug |
| Drug |
0.5/99.5 |
0.75/99.25 |
| No Drug |
24.75/75.25 |
49.5/50.5 |
As we expected, A is much worse off as a result of drug
testing. What's worse, B is better off doping because s/he can't be
found out! The winning strategy is for A not to dope, but for
B to dope. If we had tested just B instead, then the entries of the
payoff matrix will simply flip.
Solution part 5: The
answer is simply the average of the above payoff matrix (for
testing and its flipped version (for testing B). We get the
following table
| A/B |
Drug |
No Drug |
| Drug |
50/50 |
50/50 |
| No Drug |
50/50 |
50/50 |
Now probablistically it makes no difference whether an
athletes dopes or not. In reality being discovered of doping will
severly damage the reputation of an athlete. So both athletes will
choose not to dope. This scheme only requires one testing per pair.
We've accomplished the same goal while saving the cost of drug
testing!