Drug Testing

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This problem involves looking at drug testing and the payoff this might give to athletes.

Imagine a drug test that is 99% accurate.

That is, if you are drug-free, there's a 99% chance you'll pass the test, and if you have taken the drug, there's a 99% chance you'll fail the test.

In addition, imagine we know that 99% of athletes DO NOT take the drug.

If an athlete is tested and fails the test, what is the probability that they have taken the drug?

Dave and Joe are athletes at approximately  the same skill level - each has an equal chance of winning in a race between the two.

If Dave takes the drug but Joe doesn't, Dave's chance of winning increases to 75%.

If Joe takes the drug but Dave doesn't, Joe's chance of winning increases to 75%.

If they both take the drug, then each has an equal chance of winning again.

Here is a payoff matrix, showing the chances of winning:
 

The payoff of taking the drug is always better than not taking the drug, so the best strategy for both athletes is to use the drug!

The race officials decide to use drug testing, so that athletes who take drugs can be disqualified.

How does the payoff matrix change if they drug test both Dave and Joe?

How does the payoff matrix change if they only drug test Dave?  

What drug testing regime do you think would be the fairest? Are there any practical issues arising from your suggestion?