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Best of Alan Krigman
What's the Expected Value of your Dollar in the Casino?18 October 2006
Are they? Customers wouldn't rush to banks that charge a dollar to draw $0.96, $0.98, or even $0.99 from their accounts. But they flock to casinos that do much the same thing, on the average. Of course, nobody takes expected value, or anything else for that matter, literally at a casino. In particular, nobody expects an average casino visit. Everybody gambles on beating the average.
It helps to understand, though, that winners don't quite balance losers. This,
because the friendly folks who run the fun take something from the till for
their trouble. The smaller this take, the greater the expected value of the
money bet. And the more the cumulative rewards reaped by gamblers who succeed
compared with the combined penalties paid by those who fail.
If you know the chance of winning a proposition and the payoff per dollar bet, it's easy to find expected value. Just multiply the probability times the "return" - the total you get back the original bet plus the payoff. With everything on a "per dollar" basis, the return is the payoff plus one. In situations like Place bets at craps, which pay in proportions like $7 to $5 on the five or nine, there's an extra step involved. Divide the payoff by the size of the bet to get the "per dollar" figure; for the five or nine, payoff would be $7/5 or $1.40 per dollar.
The accompanying table gives expected values of a dollar bet with the corresponding probabilities and payoffs for some typical cases. Note that neither chance nor payoff alone predict whether expected value is high or low. For instance, the two double zero roulette bets have the same expected value, even though one is 12 times tougher to win and the other pays 17.5 times more.
Expected value of $1 bet on some typical propositions
Some table games offer real, not pie-in-the-sky, opportunities to do exactly this. To maximize expected value. In blackjack, it's what Basic Strategy is all about. At craps, ignoring the contentious question of dice control, you can avoid everything but Pass, Come, Don't Pass, and Don't Come while taking or laying the maximum allowable Odds. At baccarat, you can forget patterns and bet solely on Banker rather than Player (and surely never on Tie). Solid citizens who aren't doing these sorts of things when they can, even when they win in a given session, are dumping dollars into devices that downgrade them deeper than the casino demands for a shot at a profit. Presumably, they haven't learned the lesson of this lyric by the poet, Sumner A Ingmark:
Dollars the best part of boomerangs lack,
Best of Alan Krigman