The Science of Discworld Revised Edition

Grand Prix circuit, last race of the 1997–98 Formula One motor racing season … Ace driver Michael Schumacher is one Championship point ahead of arch-rival Jacques Villeneuve. Villeneuve’s team-mate Heinz-Harold Frentzen may well play a crucial tactical role. The drivers are competing for ‘pole position’ on the starting grid, which goes to whoever produces the fastest lap in the qualifying sessions. So what happens? Unprecedentedly, Villeneuve, Schumacher, and Frentzen all lap in 1 minute 21.072 seconds, the same time to a thousandth of a second. An amazing coincidence.
    Well: ‘coincidence’ it surely was – the lap times
coincided
. But was it truly amazing?
    Questions like this arise in science, too, and they’re important. How significant is a statistical cluster of leukaemia cases near a nuclear installation? Does a strong correlation between lung cancer and having a smoker in the family really indicate that secondary smoking is dangerous? Are sexually abnormal fish a sign of oestrogen-like chemicals in our water supply?
    Another case in point. It is said that 84% of the children of Israeli fighter pilots are girls. What is it about the life of a fighter pilot that produces such a predominance of daughters? Could an answer lead to a breakthrough in choosing the sex of your children? Or is it just a statistical freak? It’s not so easy to decide. Gut feelings are worse than useless, because human beings have a rather poor intuition for random events. Many people believe that lottery numbers that have so far been neglected are more likely to come up in future. But the lottery machine has no ‘memory’ – its future is independent of its past. Those coloured plastic balls
do not know
how often they have come up in previous draws, and they have no tendency to compensate for past imbalances.
    Our intuition goes even further astray when it comes to coincidences. You go to the swimming baths, and the guy behind the counter pulls a key at random from a drawer. You arrive in the changing room and are relieved to find that very few lockers are in use … and then it turns out that three people have been given lockers next to yours, and it’s all ‘sorry!’ and banging locker doors together. Or you are in Hawaii, for the only time in your life … and you bump into the Hungarian you worked with at Harvard. Or you’re on honeymoon camping in a remote part of Ireland … and you and your new wife meet your Head of Department and
his
new wife, walking the other way along an otherwise deserted beach. All of these have happened to Jack.
    Why do we find coincidences so striking? Because we expect random events to be evenly distributed, so statistical clumps surprise us. We think that a ‘typical’ lottery draw is something like 5, 14, 27, 36, 39,45, but that 1, 2, 3,19, 20, 21 is far less likely. Actually, these two sets of numbers have exactly the same probability, which for the UK lottery is: 1 in 13,983,816. A typical lottery draw often includes several numbers close together, because sequences of six random numbers between 1 and 49, which is how the UK lottery works, are more likely to be clumpy than not.
    How do we know this? Probability theorists tackle such questions using ‘sample spaces’ – their name for what we earlier called a ‘phase space’, a conceptual ‘space’ that organizes all the possibilities. A sample space contains not just the event that concerns us, but all possible alternatives. If we are rolling a die, for instance, then the sample space is 1, 2, 3, 4, 5, 6. For the lottery, the sample space is the set of all sequences of six different numbers between 1 and 49. A numerical value is assigned to each event in the sample space, called its ‘probability’, and this corresponds to how likely that event is to happen. For fair dice each value is equally likely, with a probability of 1/6. Ditto for the lottery, but now with a probability of 1 /13,983,816.
    We can use a sample space approach to get a ball-park estimate of how amazing the Formula One coincidence was. Top drivers all lap at very nearly the same speed, so the three fastest times can easily fall inside the same tenth-of-a-second period. At intervals of a thousandth of a second, there are one hundred possible lap times for each to ‘choose’ from: this list determines the sample space. The probability of the coincidence turns out to be one chance in ten thousand. Unlikely enough to be striking, but not
so
unlikely