Ergos

Rod Liddle in this week's Spectator magazine writes a piece on Monty Hall. It's a sort of card game. You place three cards face down on a table; two of the cards are jokers, one is an ace. You ask a fellow player to put his finger on the card he thinks might be the ace. He chooses a card. You pick up the other two one of which is a joker. You show him a joker and then place the other card face down next to the card he has already chosen. You now ask him if he wishes to change his mind and switch to the card you have just put down. Almost inevitably, says Rod Liddle, he doesn't change his mind. But he should because he'd have a better chance of picking the ace.
A similar "game" was played on an American gameshow in which there were three doors behind which were a car and two goats. A contestant was asked to choose a door to win the car. The compere, knowing where the two goats were, would open a door to show a goat then asking if the contestant wanted to change their original choice. Usually they didn't. When the compere announced that he should have changed so that he'd have had a better chance of winning, there was uproar from the country's mathematicians. Even the famous mathematician Paul Ergos said it was a fifty/fifty chance, no greater than that.
He was wrong. There is a bigger chance if you change your choice. Why?
The explanation is this: when you first choose you have a 1 in 3 chance of being correct; when you decide not to change you choice, you have a 1 in 2 chance which, of course, is higher.
Ergos had to have it demonstrated on a computer which played the game many thousands of times.
Ergos was a strange guy. He travelled round the place going to university maths departments, picking brains, doing calculations; when he had, so to speak, absorbed all the knowledge he wanted, he'd leave and go to another university. He once won a prize of a few thousand pounds which he gave to a beggar he passed on the street. When he died a few years back I read his obituary in The Times after which I said to my wife "There's an interesting fellow: mathematician." "Interesting? Why?" "Well," I said, "he showed....." here I can't recall the exact details of the work he did but it had to do with prime numbers - nothing to do with life outside the abstract world of mathematics. "What good is that?" she said. "What good is music?" I replied. "It's pleasant to listen to." "Maths is pleasant to do," I replied.
I'm afraid not many people believe that.