Probability is one of those things we all get wrong鈥� deeply wrong. The good news is we鈥檙e not the only ones, says , a mathematician at the University of Sussex in Brighton, UK, and author of . 鈥淢any pure mathematicians claim that probability has many unreasonable answers.鈥�
Take the classic problem of a class of 25 schoolchildren. How likely is it that two of them share the same birthday? The common-sense answer is that it is not implausible, but quite unlikely. Wrong: it鈥檚 actually just under 57 per cent.
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Or the celebrated Monty Hall problem, named after the former host of US television game show Let鈥檚 Make a Deal. You鈥檙e playing a game in which there are three doors, one hiding a car, two of them goats (see illustration). You choose one door; the host of the game then opens another, revealing a goat. Assuming you鈥檇 rather win a car than a goat, should you stick with your choice or swap?
The naive answer is it doesn鈥檛 matter: you now have a 50-50 chance of striking lucky with your original door. Wrong again.
But if probability makes even experts grumble, how do we get it right? Simple, says mathematician Ian Stewart of the University of Warwick in the UK: do things the hard way. 鈥淭he important thing with probability is not to intuit it,鈥� he says. Think carefully about how the problem is posed and do your sums diligently, and you鈥檒l arrive at the right answer 鈥� eventually.
With the birthday problem, the starting point is to realise that you鈥檙e not interested in individual schoolchildren, but pairs. In a class of 25, there are 300 pairs to consider and in most years 365 days on which each might share a birthday. Factor all that in, and you end up crunching some truly astronomical numbers to arrive at the answer. 鈥淎ny coincidence like that is remarkable in itself, but when you ask how many times it would happen, that number is so vast it鈥檚 not remarkable at all,鈥� says Haigh.
With the , meanwhile, the chance you chose the right door in the first place is 1/3 鈥� and that doesn鈥檛 change whatever happens afterwards. Since the host has revealed a goat, there is now a 2/3 probability that the car is behind the other door 鈥� and you are better off swapping.
There are a few caveats: if the host is so devious as only to open a door if you chose the right one in the first place, you鈥檇 be mad to swap. Ditto if you want a goat rather than the car. That illustrates another important rule in thinking about probability, says Haigh. 鈥淚t is very important to know your assumptions. Very subtle changes can change the outcome.鈥�
All this is very well when the boundaries of the problem are clear and the possible outcomes quantifiable. Toss a fair coin and you know you have a 50 per cent chance of heads 鈥� because you can repeat the exercise over and over again if necessary.
But what about a 50 per cent chance of rain today, or of a horse with even odds winning a race? No amount of expert advice can help us assess the true worth of such 鈥渟ubjective鈥� probabilities, which are fluid and often based on inscrutable expertise or complex modelling of an unpredictable world. Sometimes you do just have to go with your gut instinct 鈥� and be prepared to be wrong.
Read more: 鈥�Get your head around the 13 boldest ideas in science鈥�
This article appeared in print under the headline 鈥淗ow to think about probability鈥�
