The use of performance enhancing drugs is the dark side of sport, and there are stringent measures in place to try to screen athletes. But what's the likelihood that an athlete who fails a drug test has actually taken the banned drug? What do you think is the fairest way to construct a drug-testing regime? This activity, aimed at A-level students (Key Stage 5), explores some of the mathematics involved, inviting students to work with conditional probabilities and introducing the payoff matrix, an important representation in Game Theory.

Detailed teachers' notes for this activity are available on our NRICH website.

##### Stage: 5 Challenge Level:

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:

Dave/Joe (%) | Drug | No Drug |

Drug | 50/50 | 75/25 |

No Drug | 25/75 | 50/50 |

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? **

**How does the payoff matrix change if they randomly drug test either Dave or Joe with a 50% chance?**

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

**Further information and related resources**

Detailed teachers' notes for this activity are available on our NRICH website.

Our article The logic of drug testing by John Haigh and Mike Pearson examines conditional probability in sports doping tests, and includes an animated illustration.

Our article Blast it like Beckham by John Haigh also examines payoff matrices, looking at what Game Theory can tell us about how to take a penalty kick in football.