EXAMPLE 26 The Casino Gets Rich

The casino bets with all its customers—perhaps 100,000 individual red/black roulette bets in a week. The central limit theorem guarantees that the distribution of average customer winnings on 100,000 bets is very close to normal. The mean, from the gambler’s point of view, is still the mean outcome for one bet, −$0.053, a loss of 5.3 cents per dollar bet. The key point is that the standard deviation is much smaller when we average over 100,000 bets. It is

Here is what the 99.7% confidence interval estimate of the average result looks like after 100,000 bets:

Because the casino covers so many bets, the standard deviation of the average winnings per bet becomes very small. Not only is the mean negative, but the entire 99.7% confidence interval is also in the negative region, so the total result is virtually certain to be in the casino’s favor. The gamblers’ losses and the casino’s winnings are almost certain to average between 4.4 and 6.2 cents for every dollar bet.

The gamblers who collectively place those 100,000 bets will lose money. The probable window of their losses is

The gamblers are almost certain to lose—and the casino is almost certain to collect—between $4400 and $6200 on those 100,000 bets. What’s more, the interval of average outcomes continues to narrow as still more bets are made. That is how a casino can make a business out of gambling.