### Rain or Shine

Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.

### Squash

If the score is 8-8 do I have more chance of winning if the winner is the first to reach 9 points or the first to reach 10 points?

### Snooker

A player has probability 0.4 of winning a single game. What is his probability of winning a 'best of 15 games' tournament?

# Drug Testing

##### Stage: 5 Challenge Level:

For the first part:

If test is positive, we are 99% sure of correctness
If test is negative, we are 99% sure of correctness

99% of athletes DO NOT take the drug. Which means 1% do.

1% of 99% who have not actually taken the drug will test positive (false positive).

99% of 1% who have taken the drug will test positive (true positive).

Total positive tests = false positives + true positives = $0.01 \times 0.99 + 0.99 \times 0.01 = 0.0198 = 1.98%$ So 1.98% of tests are positive.

Of these, 0.99% are false positives and 0.99% are true positives, so the probability an athlete has taken the drug given that they tested positive is 50%.

If both athletes are drug tested:

If both athletes take the drug, then each has 1%*(100%-1%*50%)=0.95% chance of winning if he luckily passes the test, unless the other athlete also passes and beats him.

If neither takes the drug, 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 takes the drug and the other doesn't, then the drug-taking 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

 Dave/Joe (%) Drug No Drug Drug 0.95/0.95 0.7525/98.2575 No Drug 98.2575/0.7525 49.995/49.995

If only Dave is tested:
If both athletes take the drug, then Dave has 1%*50% = 0.5% chance of winning. Since Joe cannot be disqualified, he will have 1-0.5%=99.5% chance of winning. If only Dave takes the drug, then he has 1%*75%=0.75% chance of winning. If only Joe takes the drug, then Dave has 99%*25% = 24.75% chance of winning. If neither dopes, then Dave has 99%*50% chance of winning.

 Dave/Joe (%) Drug No Drug Drug 0.5/99.5 0.75/99.25 No Drug 24.75/75.25 49.5/50.5

Dave is much worse off as a result of drug testing. What's worse, Joe is better off taking the drug because he can't be found out!  The winning strategy is for Dave not to take the drug, but for Joe to take it. If we had tested just Joe instead, then the entries of the payoff matrix will simply flip.

If Dave or Joe are tested randomly, with probability 50%
The answer is simply the average of the above payoff matrix (for testing Dave) and its flipped version (for testing Joe). We get the following table

 A/B Drug No Drug Drug 50/50 50/50 No Drug 50/50 50/50

Now it makes no difference whether an athlete takes the drug or not. In reality being caught drug-taking will severly damage the reputation of an athlete (and possibly lead to being banned from competing). So both athletes will choose not to take the drug.