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Games of Chance and the Law of Averages

A game of chance is like flipping a coin or spinning a wheel with ten numbers. What happens is what happens. A player can guess what the outcome will be but cannot influence it. Games of chance operate according to the law of averages If you have a fair coin and flip it ten times, the law of averages leads you to expect that approximately half of the tosses will come up heads and the other half tails. If a roulette wheel has 38 slots, the law of average suggests that the ball will fall into a particular slot one time in 38 spins.

The coin, the roulette ball, and the dice, however, have no memory. They just keep plugging along doing their thing. If I toss a coin and come up with heads nine times in a row, what are my chances of getting heads on the tenth toss? The answer is 50%, the same chance as getting heads on any toss. Each toss is completely independent of any other toss. When the coin goes up in the air that tenth time, it doesn't know that tails has not come up for a while, and certainly has no obligation to try to get the law of averages back into whack.

Though most gamblers are familiar with the law of averages, not all of them understand how it works. The operative word, as it turns out, is "averages," not "law." If you flip a coin a million times, there is nothing that says you will get 500,000 heads and 500,000 tails, no more than there is any assurance you will get five heads and five tails if you flip a coin ten times. What the law of averages does say is that, in percentage terms, the more times you toss the coin, the closer you will come to approximating the predicted average.

If you tossed a coin ten times, for example, you would not be surprised to get six tails and four heads. Six tails is only one flip off the five tails and five heads that the law of averages tells you is the probable outcome. By percentage, however, tails came up 60% (six of ten) of the time, while heads only came up 40% (four of ten) of the time. If you continued flipping the coin for a million tries, would you be surprised to get 503, 750 tails and only 496 250 heads, a difference of 7,500 more tails than heads? The law of averages stipulates that the more we toss (and a million tosses are certainly a lot more than ten tosses) the closer we should come to approximating the average, but here we are with a huge difference of 7,500 more tails. What went wrong?

Nothing went wrong. True. after ten flips, we had only two more tails than heads, while after a million flips we had 7,500 more tails than heads. But in terms of percentage, 503,750 tails is 50.375% of one million, only about one-third of a measly percent from what the law of averages predicts. The law of averages is about percentages. Gambling is about dollars out of your pocket. If you had bet a dollar on heads each toss, you would have lost $2 after ten flips. After a million flips you would have lost $7,500. The law of averages behaved just a mathematical theory predicted, but that's probably not much consolation for going home broke.


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