Thinking about expected values

As with probability, it is worth exploring a few fine points about expected values and the law of large numbers.

How large is a large number? The law of large numbers says that the actual average outcome of many trials gets closer to the expected value as more trials are made. It doesn’t say how many trials are needed to guarantee an average outcome close to the expected value. That depends on the variability of the random outcomes.

The more variable the outcomes, the more trials are needed to ensure that the mean outcome is close to the expected value. Games of chance must be quite variable if they are to hold the interest of gamblers. Even a long evening in a casino has an unpredictable outcome. Gambles with extremely variable outcomes, like state lottos with their very large but very improbable jackpots, require impossibly large numbers of trials to ensure that the average outcome is close to the expected value. (The state doesn’t rely on the law of large numbers—most lotto payoffs, unlike casino games, use the pari-mutuel system. In a pari-mutuel system, payoffs and payoff odds are determined by the actual amounts bet. In state lottos, for example, the payoffs are determined by the total amount bet after the state removes its share. In horse racing, payoff odds are determined by the relative amounts bet on the different horses.)

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Though most forms of gambling are less variable than the lotto, the practical answer to the applicability of the law of large numbers is that the expected value of the winnings for the house is positive and the house plays often enough to rely on it. Your problem is that the expected value of your winnings is negative. As a group, gamblers play as often as the house. Because their expected value is negative, as a group they lose money over time. However, this loss is not spread evenly among the many individual gamblers. Some win big, some lose big, and some break even. Much of the psychological allure of gambling is its unpredictability for the player. The business of gambling rests on the fact that the result is not unpredictable for the house.

STATISTICAL CONTROVERSIES

The State of Legalized Gambling

Most voters think that some forms of gambling should be legal, and the majority has its way: lotteries and casinos are common both in the United States and in other nations. The arguments in favor of allowing gambling are straightforward. Many people find betting entertaining and are willing to lose a bit of money in exchange for some excitement. Gambling doesn’t harm other people, at least not directly. A democracy should allow entertainments that a majority supports and that don’t do harm. State lotteries raise money for good causes such as education and are a kind of voluntary tax that no one is forced to pay.

These are some of the arguments for legalized gambling. What are some of the arguments against legalized gambling? Ask yourself, from which socioeconomic class do people who tend to play the lottery come, and hence who bears the burden for this “voluntary tax”? For more information, see the sources listed in the Notes and Data Sources at the end of this chapter.

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Is there a winning system? Serious gamblers often follow a system of betting in which the amount bet on each play depends on the outcome of previous plays. You might, for example, double your bet on each spin of the roulette wheel until you win—or, of course, until your fortune is exhausted. Such a system tries to take advantage of the fact that you have a memory even though the roulette wheel does not. Can you beat the odds with a system? No. Mathematicians have established a stronger version of the law of large numbers that says that if you do not have an infinite fortune to gamble with, your average winnings (the expected value) remain the same as long as successive trials of the game (such as spins of the roulette wheel) are independent. Sorry.