The Art of Statistics: Learning from Data

by David Spiegelhalter

This is the idea of ‘expected frequency’. When faced with the problem of the two coins, you ask yourself, ‘What would I expect to happen if I tried the experiment a number of times?’ Let’s say that you tried flipping first one coin, and then another, a total of four times. I suspect that even a politician could, with a bit of thought, conclude that they would expect to get the results shown in Figure 8.2. So 1 in 4 times you would expect to get two heads. Therefore, the reasoning goes, the probability that on a particular attempt you would get two heads is 1 in 4, or ¼. Which, fortunately, is the correct answer. This expected frequency tree can be transformed into a ‘probability tree’ by labelling each ‘split’ with the fraction of occasions that it is taken (Figure 8.3). It should then be clear that the overall probability for an entire ‘branch’ of the tree, say a head followed by a head, is obtained by multiplying the fractions on the splits along the branch, so that ½ × ½ = ¼. Probability trees are a widespread and extremely effective way of teaching probability in school. Indeed, we can use this simple example of flipping two coins to see all the rules of probability, since the probability tree shows that The probability of an event is a number between 0 and 1: 0 for impossible events (for example, flip no heads and no tails), 1 for certain events (flip any of the four possible combinations). Complement rule: the probability of an event happening is one minus the probab...

Probablity tress are like decision trees