Tuesday, January 17, 2006

On probability

You are a contestant in a game show. You choose from three closed boxes. One contains the keys to a car and the other two a picture of a goat. You make a choice. The host opens one of the other doors to reveal – a goat. He taunts you to change the decision. Should you switch to the other closed box?

This is the Monty Hall problem. When the problem was first published in an American magazine in the 1970s, thousands of readers — including professors of statistics — alleged an error.

Consider another problem. One person in a 1,000 suffers from a rare disease. You have been tested randomly for this illness and tested positive. The test gives a correct diagnosis in 99 per cent of cases. How likely is it that you have the disease? If you’re much stressed then don’t be. It’s not very likely that you have the disease. In random groups of 1,000 people an average of 10 would display false positives and only one would be correctly diagnosed with the disease.

Suppose that two editors, Amar and Akbar, independently check your manuscript. Amar finds 3 mistakes and Akbar finds 4, and 2 of the mistakes are commonly found. Can you see that we would expect only one unfound error remaining?

These are just some examples of probability problems I found hard. But then, as Stephen Jay Gould said: The human mind did not evolve to deal with probabilities.

And if you’re still scratching your head, the answer to the Monty Hall problem is, yes you should switch.