You don't have to watch sports coverage very long before you hear about the supposed law of averages. I heard it cited this weekend as an explanation of why Tiger Woods failed to win the U. S. Open.

The idea was that since Mr. Woods had won four major tournaments in a row, he was unlikely to win a fifth. The fact is that the number of tournaments he has won in the past does not affect the probability of his winning this week's tournament. The probability remains the same regardless of what he has done in the past.

The simple fact is Mr. Woods is unlikely to win every tournament. The probability that he will win any single tournament is less than 1. There is no need to invoke a law to explain why he didn't win.

In statistics, the preferred term for the law of averages is the gambler's fallacy. It is the belief that independent events are not independent, and that an increase in the percentage of times an event happens implies that its probability of happening again is reduced. For example, if red has won five times in a row at the roulette table, a gambler might bet black because red is now less likely to win. Red is, in fact, as likely to win as it ever was (assuming, of course, the wheel is honest).

The gambler's fallacy is probably often due to the making of incorrect comparisons. Once Tiger Woods had won four major tournaments in a row, he was indeed unlikely to win the next four. However, he would have been just as unlikely to win the next four as he would have been if he had never won a major tournament in his life. In particular, he would have been just as likely to win the next tournament.

This fallacy is also probably often due to careless thinking about regression to the mean. People who produce extremely high performance on one occasion tend not to perform as well on the next occasion because they don't have as much luck. Extreme performance tends to be due in larger part to error (also known as luck) than is less extreme performance.

However, regression to the mean does not imply that the probability of performing that well has been reduced. If a scratch golfer completes a round at six under par he is indeed unlikely to do as well the next time. The probability that he will do as well, though, has not changed.

Law and Fallacy © 2001, John FitzGerald

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