Drug testing
Problem
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?
What drug testing regime do you think would be the fairest? Are there any practical issues arising from your suggestion?
Getting Started
It may be helpful to consider an ordering of the events, with the associated probabilities at each stage.
For example, first the athletes decide whether to take the drug or not. Then the race takes place. Then the drug testing takes place.
Student Solutions
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
|
| Dave/Joe (%) | Drug | No Drug |
| Drug | 0.5/99.5 | 0.75/99.25 |
| No Drug | 24.75/75.25 | 49.5/50.5 |
| A/B | Drug | No Drug |
| Drug | 50/50 | 50/50 |
| No Drug | 50/50 | 50/50 |
Teachers' Resources
Why do this problem?
This problem invites students to work with some tricky conditional probabilities in the context of drug testing in sport. It also introduces the payoff matrix, an important representation in Game Theory.Possible approach
Once students have had a chance to engage with and work on this first question, introduce the second part of the problem. Give students time to make sense of the payoff matrix, and then split the class into two groups, with half the students working on the question "How does the payoff matrix change if they drug test both Dave and Joe?" and the other half working on "How does the payoff matrix change if they only drug test Dave?"
After students have had time to tackle their question, bring the class back together and invite each group to present their working out and the payoff matrix.
Finally, challenge students to answer the question "How does the payoff matrix change if they randomly drug test either Dave or Joe with a 50% chance?" and then take some time to discuss the pros and cons of different drug-testing regimes.