Magazine article The Spectator

Monty Hall Will Change the Way You Think

Magazine article The Spectator

Monty Hall Will Change the Way You Think

Article excerpt

Rod Liddle on a curious and startling mathematical conundrum that demonstrates how easily we are led astray by bad logic and unreliable intuition

Here's a game to play this evening with your wife or your catamite. It is an incredibly boring game, but it will help you understand the world better than a bunch of Nobel prizewinners and more than 100 mathematical geniuses, who we will come to in good time.

Take three cards - an ace and a couple of jokers. Shuffle them up. Lay the cards face down in front of your partner and tell her that if she picks the ace, you'll give her a bourbon or maybe a garibaldi biscuit. If she picks one of the jokers, however, she gets nowt. Tell her not to turn over the card of her choice just yet, simply to tap it. When she's done that, pick up the two other cards. At least one of them will be a joker - reveal this card to your missus and put the other card, undisclosed, back in front of her. Now ask her if she wishes to switch from her original choice. Make a note on a piece of paper of her decision, and also whether she wins the biscuit or not. Repeat this entire procedure about 100 or better still 1,000 times - I told you, it's boring. Make sure you have lots of biscuits ready - if she switches every time.

Most people don't switch, however, so the likelihood is you will need far fewer biscuits.

This is the Monty Hall dilemma and it was back in the news last week, picked up in the weekend's broadsheets, and has been rolling around the blogs ever since. It resurfaces once in a while and everybody is always dutifully astonished, not to mention utterly disbelieving whenever it emerges in one or another guise. On this latest occasion it was presented in a slightly different (although pretty familiar) form, at the start of an article by Alex Bellos in New Scientist.

He was quoting a puzzle designer called Gary Foshee who was addressing a symposium in Atlanta and who began his speech with this conundrum: 'I have two children.

One is a boy born on Tuesday. What is the probability that I have two boys?' This is the Monty Hall dilemma, and indeed my card puzzle, slightly rewritten - the same sort of principles apply. In this case, bizarrely, the information that the boy was born on a Tuesday (or any other day of the week) is crucial to the calculation and not, as you might imagine, a red herring. Without that information, you can calculate the odds by looking at the combinations of two children it is possible to have - (gg, gb, bg, bb). As we already know that one child is a boy we can eliminate (gg) - and therefore the odds of two boys are one in three. When you add in the information that the boy was born on a Tuesday, however, the probability changes from one in three to 13/27, or almost 50 per cent - a huge difference. Once you have listed the equally likely possibilities of children, together with days of the week, you end up with 27 possible permutations, 13 of which are two boys.

Now, I think this is a less startling outcome than my card trick or the Monty Hall dilemma - which I promise I'll come to - because when you strip it down it is not hugely counterintuitive. By which I mean that if I have one boy and asked you what the likelihood of my next child being a boy (ignoring all genetic factors) you would probably say 50 per cent - which would not be far wrong, as it turned out. …

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