The False-Positive Paradox as a Class Activity/Discussion Point

There is a test that screens for some rare disease (only 1 in every million people is infected). It is 99% accurate (it gives the correct answer of positive or negative 99% of the time). You go for the test and to your dismay it returns a positive result!
What is the chance that you actually have the disease?

You may think that its pretty likely you have the disease. However, from the information above, the actual probability that you have the disease is still very unlikely (1 in 10'102). This is the basic premise of the false-positive paradox.

I don't really like that it is named as a paradox, because it is not actually in any way paradoxical, just counter-intuitive. It works because there are so many more non-infected people (999'999) that each have a 1% chance of getting a false positive than the number of infected people (1) who has a 99% chance of getting a true positive result.

This counter-intuition can cause exactly the sort of cognitive-disonance that can be so useful in getting people to realise that their current understanding needs improving, but in its current form its pretty wordy, inaccessible and abstract. I wanted to make it in to a class activity. Instructions below.

 

 

 

How to run the Class Activity/Discussion Point:

• As students enter the room, give them each a slip of paper with a random word on it. Tell them not to show anyone else.
• Once they're settled, tell them that there is a (make-believe) disease going round school that only affects about one student per class. In this class the one student, the one with the word banana on their slip of paper, is infected (Don't tell anyone if its you or not!).
• Tell them you have a way of checking if anyone has the disease, but its not always accurate. To perform the test, you will ask the student if they are infected. Before answering, the student will roll a dice and will tell the truth if they roll a 1, 2, 3, 4 or 5, but they will lie if they roll a 6.
• Perform this test on some students until someone claims to be infected.
• Ask the class how likely is it that this student is actually infected. Ask for justifications. Most will say very likely, some may evan say 5/6.
• Perform the test on a few more students until you get a few more claiming to be infected.
• But they can't all be 'very likely'! What is going on here? 

By the way, make sure the students know the disease is not real!

Would this work in your classroom? Could it be improved? At the moment its just an idea and I'd like to know what you think.