Shifting probabilities

One of the most difficult ideas for students learning about probability in the early stages is that the probability of something happening changes when we are given different evidence. This reflects the fact that when we get more evidence we change our perception of how likely it is that something will or will not happen.

A simple example is when we toss two coins. What is the probability that they both turn up heads? Since this is one of four possible outcomes (HH, HT, TH, TT), the probability is reckoned to be 1/4. However if I tell you that one of the coins lands as a head, then there are now only three possible outcomes (HH, HT, TH) so the probability of two heads is now 1/3. The new evidence changes the probability. If I then tell you that one of the coins is a 10p piece and the other is a 20p piece, and the 10p piece has turned up head, there are now only two possible outcomes: the 20p piece might be a head or it might be a tail. So the probability of both being heads is now 1/2.

There is a great example of this being discussed on various blogs at present – and that was also featured in a recent 'More or Less' broadcast on Radio 4. A man tells you that he has two children and one of them is a boy, the question is what is the probability that the other one is a boy? That's pretty straightforward: 1/3, because he could have BB, BG or GB. (Compare the coins question above). But if he tells you that he has two children and one of them is a boy who was born on a Tuesday, that actually changes the probability that both are boys. Or that is at least what some mathematicians are proposing. This is wonderfully subtle stuff! For some of the discussion, start at blog.tanyakhovanova.com/?p=221.