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### Number and algebra

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# Drug Testing

### 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.

### Key questions

### Possible extension

Invite students to consider the possible outcomes when three athletes compete for Gold, Silver and Bronze, with different drug testing regimes.

### Possible support

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

Introduce the first part of the problem:

"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 tested and fails the test, what is the probability that they have taken the drug?"

You may wish to show students this animation to introduce Bayes Theorem together with a visual way of thinking about conditional probability in circumstances such as this. This Plus article also examines conditional probability in the context of sports doping, including
a tailored version of the animation.

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.

Imagine Dave takes the drug and Joe doesn't, and they are both tested. What are the different possible outcomes?

What is the probability of each outcome?

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.

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.

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?

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