Last night I had the pleasure to meet another of my mathematical heroes: Ian Stewart. Ian holds a special place in my little sheepy heart because it was through reading his books that I first became aware of the beautiful subject that is mathematics. I was a bit nervous when I went up to shake his hand, and indeed he viewed me at first with a strange mixture of puzzlement and incredulity. So I decided to break the ice with a mathematical sheep joke:
An astronomer, a physicist and a mathematician are on a train in Scotland. The astronomer looks out of the window, sees a black sheep standing in a field, and remarks, “How odd. Scottish sheep are black.” “No, no, no!” says the physicist. “Only some Scottish sheep are black.” The mathematician rolls his eyes at his companions’ muddled thinking and says, “In Scotland, there is at least one sheep, at least one side of which looks black.”
Ian was at the Science Festival to tell us about his new book Cows in the Maze, which is another collection of puzzles from Ian’s column in Scientific American. (The other two books being Math Hysteria and How to Cut a Cake.) I wouldn’t recommend Googling for “Cows in the Maze” because, like me, you may end up wasting 3 hours of your life playing a Flash game trying to get cows out of a maze. However, the book itself is certainly worth a look as it contains some interesting mathematical curiosities, touching topics as diverse as time travel, teardrops, courtrooms, knots (yay!) and, of course, cows.
The problem which most captured my attention last night was the question of whether a confession of guilt may actually add more weight to a defendant’s likelihood of innocence. To illustrate the principle behind the theory, Ian first gave us a much simpler question: “If a family has two children and one of the children is a girl, what is the probability of the other child being a girl?” At first glance, the fact that one child is a daughter doesn’t change the outcome of the gender of the second child, so it should be 50/50. Furthermore, if you are given information about whether the girl child is older or younger, the probability of the second child being a girl is 50/50. Ok, but the girl child has to be either older or younger! And yet, if you are not given this extra information, the probability of the second child being a girl is 1 in 3. Eh?
Here’s the explanation. If you have two children, there are usually 4 choices of gender matches: BB, BG, GB or GG. But once you are told that one child is a girl, this narrows to three choices: BG, GB or GG. And only one of these three options is girl-girl. However, if you are told that the girl is the older child then you are only left with two choices: GB and GG, so there’s a 50% chance of girl-girl this time. Similarly if the girl is younger. Weird, huh?
It’s just one of many examples of how our intuition about probability can fail us. (For an even more counterintuitive puzzle, you can ask “If a family has two children and one of the children is a boy that was born on Tuesday, what is the probability of the other child being a boy?” See Alex Bellos’ website for the answer and explanation!)
How does this relate to a defendant in a courtroom? Well, suppose you are halfway through a trial and the evidence so far gives a particular probability of the guy being guilty. You are then told that the defendant has confessed to being guilty. Does this increase or decrease his probability of guilt? Certainly the extra information will change the probabilities involved. Interestingly, the probability of guilt will only increase if an innocent person is less likely to confess than a guilty one. In many cases this is unlikely to be true, since a hardened criminal may have been trained to resist interrogation, whilst a poor innocent person may quickly confess in order to get out of the traumatising situation of the courtroom.
Lots of food for thought there! Speaking of food, I’m off to find some tasty grass…