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What's the Expected Value of your Dollar in the Casino?

18 October 2006

By Alan Krigman

Casinos often tout the "return" or "payback" percentages delivered by their slot machines. You're encouraged to believe that averages such as 96 percent are good, 98 percent better, and 99 percent gifts bestowed by the bosses in fits of philanthropy.

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.

Bets on table games are usually characterized by "edge" or "house advantage." But they can also be described using expected value. In fact, the parameters are complimentary. An edge of 2 percent is equivalent to a return of 100 - 2 or 98 percent. That is, at 2 percent edge, a dollar bet has an expected value of $0.98.

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

bet chance of winning payoff expected value
00 roulette dozen

$ 2.00

00 roulette straight-up
craps Pass, no Odds
craps five
craps 12
baccarat Player
baccarat Banker
baccarat Tie

The formula shows that, in principle, the expected value of a bet can be increased by improving the chance of winning, raising the payoff, or both. Consider the 12 at craps. Assume, arguendo, that you could set and throw dice in a manner that eliminated all possibility of tossing a two without affecting how other totals could be formed. Then the chance of a 12 would be 1/35, not 1/36. And the expected value would be (1/35) x $31 or $0.8857, not $0.8611. Alternately, if a casino restrained its rapacity and paid $31 instead of $30 for $1 on the 12, expected value would be (1/36) x $32 or $0.8889. Combining skill as a shooter and the beneficence of the bosses would yield (1/35) x $32 or $0.9143.

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,
Throw them away; don't expect they'll come back.

Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns were focused on those interested in gambling probability and statistics. He passed away in October, 2013.