Introduction to Week 2

Hello, and welcome to Week 2 of Teaching Probability. Week 1 of the course covered a lot of material,

and finished by setting you a practical challenge: running a probability activity with a suitable class. We do hope that you were able to do that. And even if you couldn’t run a class activity, we hope you tried out one of the experiments for yourself, and thought about how you would use it with a class. Of course, even if you were not in a position to run the activity you should still have a gained a lot by hearing from others about how they got on. This week’s lesson will look in more depth at some of the trickier areas of our topic, and in particular we are going to focus on teaching conditional probability.

This is a very popular topic for assessment purposes, so you may find this week’s material particularly useful if you have classes preparing for examinations. We’re going to see how the approaches that we looked at last week can be extended to these more complex problems and make them more straightforward to solve. In real life, conditional probability is often discussed in the context of cancer screening or drug testing. But these are generally not appropriate for teaching. We’ve therefore developed ‘The dog ate my homework!’, and this is the same basic structure, but it is fun rather than emotionally challenging. There are many misunderstandings with conditional probabilities. Take this headline that said that 35% of bikers were killed in road accidents.

ie, if you are a biker, the conditional probability of having a fatal accident is 35%. This is clearly nonsense! What they meant to say, and later corrected, was that 35% of fatal accidents involved bikers. ie if you’re in a fatal accident, the conditional probability of being a biker was 35%. This comes down to the choice of denominator, or ‘reference class’. The bikers in fatal accidents should be considered as a proportion of fatal accidents, not of all bikers. As with last week, we will be looking for ways to make the teaching of probability more meaningful, so that students can use and develop their existing understanding of chance.

One of the key ideas that we will be using this week is multiple representations. This is simply the idea that any probabilistic situation can be represented in a variety of ways. You’ve already seen some examples of this in Week 1. You saw situations modelled with physical apparatus and with spreadsheets, raw data in tables, and estimated probabilities. And we contrasted the use of a frequency-based approach to solve problems to the more traditional approach of using formal probabilities. We’ll take this further this week, introducing new forms of representation, that can help your students make sense of conditional probability. Be on the lookout for these as we work through this lesson.

I hope that you enjoy Week 2 of the course, and that you will find plenty of new ideas to try out, or just to make you think. We believe that the teaching of conditional probability can present a distinct set of challenges. Do you agree? Is this an area that your students find particularly challenging? Use the comments to let us know.