EXAMPLE 22 The Law of Large Numbers and the Gambling Business

The law of large numbers explains why gambling can be a business. In a casino, the house (i.e., the casino) always has the upper hand. Even when the edge is very small, such as in blackjack—where according to “The Wizard of Odds,” the house edge is only around 0.3%—the casino makes money. The house edge is defined as the ratio of the average loss to the initial bet. That means for every $10 initial bet, the gambler will lose, on average, , or 3 cents per game. Over thousands and thousands of gamblers and games, that small edge starts to generate big revenues. Unlike most gamblers, casinos are playing the long game and not just hoping for a short-term payout.

The winnings (or losses) of a gambler on a few plays are highly variable or uncertain; that’s why gambling is exciting. It is only in the long run that the mean outcome is predictable. Take, for example, roulette. An American roulette wheel has 38 slots, with numbers 1 through 36 (not in order) on alternating red and black slots and 0 and 0 on two green slots. The dealer spins the wheel and whirls a small ball in the opposite direction within the wheel. Gamblers bet on where the ball will come to rest (see Figure 8.23). One of the simplest wagers is to choose red. A bet of $1 on red pays off an additional $1 if the ball lands in a red slot. Otherwise, the player loses the $1.

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Lou bets on red. He wins if the ball stops in one of the 18 red slots. He loses if it lands in one of the 20 slots that are black or green. Because casino roulette wheels are carefully balanced so that all slots are equally likely, the probability model is

Net Outcome for Gambler

The mean outcome of a single $1 bet on red is

The law of large numbers says that the mean is the average outcome of a very large number of individual bets. In the long run, gamblers will lose (and the casino will win) an average of 5.3 cents per bet.

So the house, unlike individual gamblers, can count on the long-run regularity described by the law of large numbers. The average winnings of the house on tens of thousands of plays will be very close to the mean of the distribution of winnings. Needless to say, gambling games have mean outcomes that guarantee the house a profit; though, as we have seen, some games such as blackjack give the house a smaller advantage than others such as keno. (According to “The Wizard of Odds,” the house edge on keno is 25% to 29%!)