## Goats, buses and the lottery

### April 6, 2012

I can highly recommend Marcus du Sautoy‘s video on the Guardian’s new Newton channel, exploring the world of chance and probability.

The good professor shows how probabilities become counter-intuitive as soon as a modicum of complexity is introduced. We all know that the chance of guessing heads on a coin-flip is one-in-two, and it’s fairly straightforward to show the chance of scooping the National Lottery jackpot is just under one in 14m.

But when it comes to a game show involving goats and sports cars (known variously as the “Monty Hall” problem, after the host of the game show on which the conundrum appeared, or simply as “the goat problem”), intuition abandons us. Here’s the set-up:  there is a sports car and two goats (d0n’t ask!) behind three doors. You have to pick one. Monty then opens one of the other two doors to reveal a goat. The dilemma: Do you stick or switch to the other door?

Intuition blithely assures us nothing has changed, so it makes no difference whether you switch or not. But hold on – not so fast. Originally, the chance you’d picked the sports car was one-in-three; that’s still true. But the chance that the sports car is behind the other door is now two-in-three, much better odds. You should always switch.

In fact, the conclusion offends common sense to such an extend that (occasionally bitter) debates have sprung up about its veracity (see here and here, for instance).

(Incidentally, I had the pleasure of attending one of Marcus du Sautoy’s talks a few years back, at an Open University maths revision weekend, where he told the story of Evariste Galois, one of the most romantic figures in the history of mathematics.)

But the other point he makes in his Newton video is the crucial role sampling plays in surveys and polls – if you don’t know the sample size of a poll, you know nothing. A small random sample can provide accurate information about a very large population – but the word “random” is the key here, since achieving randomness is exceedingly tricky.

Many people assume “random” means “well spread out”, but randomness tends to clumping, which is one of the reasons why buses always come in threes. None of which, I trust, contains a goat …