Who Is Cheating?

A new test is in development to try to identify athletes who use a certain banned substance to enhance their performance.

The development team has accumulated data which indicates that although the test shows good results in detecting an athlete who has used this substance, the false positive rate is more worrying.

What is the probability that an athlete who is not taking the substance tests positive?
What is the probability that an athlete who is taking the substance tests negative?

You are going to investigate this question through a practical experiment.
You will need a die with four sides one colour (blue) and two a different colour (red), two ordinary numbered dice, and some red, blue, green and yellow multi-link cubes. (If you do not have a die with sides in two colours, use a numbered die and designate 1, 2, 3 and 4 to be blue and 5, 6 to be red).

flow chart of simulation rules

First use the flow chart to investigate what happens for one athlete (click here for a full size version or click here for a written version).

Stick your two cubes together.

What do each of these mean?

red and yellow
blue and yellow
red and green
blue and green

Now repeat the experiment 36 times in total, so that you have 36 pairs of cubes.

Are you surprised by your results?

How do they compare with what we would expect?
This worksheet will help you to work out the expected results. You need to be sure that you have correctly identified how many possible outcomes there at each stage, and how many of those give each result - this Sample Space worksheet will help you with that.

In what proportion of the 36 trials do athletes who are not taking the banned substance test positive?

Given that an athlete is not taking the banned substance, what is the probability that they test positive? (What figure should go in the denominator here?)

In what proportion of the 36 trials do athletes who are taking the banned substance test negative?

Given that an athlete is taking the banned substance, what is the probability that the test is negative? (What figure should go in the denominator here?)

This worksheet will help you to interpret the expected results as probabilities, leading to a general result for probability trees - the multiplication rule.

Extension questions

Once you have your probability tree and 2-way table, have a go at these questions:

What proportion of accused athletes were actually taking the banned substance?
What is the probability that someone who is accused is really innocent?
What is the probability that someone who tests positive is in fact taking the banned substance? (You will need to reverse your probability tree to answer this).
What is the probability that someone who tests negative is in fact taking the banned substance?

Can you suggest other questions along these lines that you can now answer?

Rules for the simulation:

Throw the coloured die.
- If it is red, the athlete is taking the banned substance. Take a red cube.
- If it is blue, the athlete is not taking the banned substance. Take a blue cube.
If the athlete is taking a banned substance (you have a red cube), throw two numbered dice.
- If the total on the two dice is 4 (ie. 1+3, 3+1, 2+2) that means a negative test result - take a green cube.
- Anything else means a positive test result - take a yellow cube.
If the athlete is not taking a banned substance (you have a blue cube), flip a coin three times.
- Three heads means a positive test result - take a yellow cube.
- Anything else means a negative test result - take a green cube.