This article is part of our collection Great Expectations: Probability through Problems.
For many students, conditional probability seems to be too hard, and pointless anyway. Frankly, who cares what the chance is of getting a blue ball, given that the previous one was red. And who wants to get tangled up with Bayes' Theorem?
If that's how it's taught, then yes, it is dry, apparently pointless, and difficult to justify spending time and effort on.
But if conditional probability is about the chance that a positive test for cancer means you actually have cancer - that matters. If it's about understanding why a simplistic approach to probability leads to miscarriages of justice - that matters. These are questions which can have life and death implications.
A psychologist, Gerd Gigerenzer, in Reckoning with Risk: Learning to Live with Uncertainty (Penguin Books, 2002) gives this example:
The chance that even a well-informed person calculates this probability correctly from information presented in this form is not high. That wouldn't matter if it were purely a mathematical problem, but failing to understand information given in this way is at the root of many medical and legal miscalculations. And no - the correct answer is not about 90% - although you could be forgiven for thinking that it is!
The same problem can be presented in this way:
These scenarios can easily be adapted to others. Teachers on a course in South Africa recently came up with these:
There is a common lesson structure for these problems. Students start by answering questions from their own results, and from the aggregated results of the whole class, using a tree diagram and 2-way table to represent their results and provide a structure to interpret the data. They then use their intuitive understanding of random events (dice and coins) to see what results they would expect, comparing the experimental data with the expected results.
The expected results, also displayed as whole numbers on a tree diagram and 2-way table, provide the data to answer questions which progress from: 'What proportion of people who experience A do we expect to experience B also?' to: 'What is the chance that a person who experiences A experiences B also?'; 'Is this the same as the chance that a person who experiences B also experienced A?' From this students answer questions of the form: 'Given that a person experiences A, what is the probability that they also experience B?' Reversing the tree diagram enables them to answer 'Given that a person experienced B, what is the probability that they also experienced A?'
Exploring all these forms of questions in this way enables students to investigate data in order to answer worthwhile questions such as 'If a person tests postive for cancer, what is the probability that they are actually suffering from cancer?' or 'If a person has two children who both die apparently from SIDS, what is the probability that this occurred by chance?'
From this, those for whom it is approriate can go onto Bayes' Theorem with a sound understanding of the basis for theory and formula.