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Best of Alan Krigman

Gaming Guru

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Playing It Smart: Probability and payoff ratio are the tails that wag the casino dog

1 April 2008

When casino gamblers grumble about house advantage or edge, they're usually thinking of the probabilities associated with a particular bet. Edge and probability are related but aren't the same. A casino can have an edge even though a solid citizen has a better chance of winning a wager. And a player could have an edge on a bet when a casino will be victor more often than vanquished. The element linking probability and edge is payoff ratio.

Here's an illustration of the first case. Bet $5 on each of two 12-number columns at double-zero roulette. You have 24 ways to win and a mere 14 to lose. That's 24 out of 38 possibilities, and 24/38 is 63.16 percent. The hitch is that $10 at risk nets $5. The edge is [(24/38)x$5 - (14/38)x$10]/$10 or 5.26 percent, the minus sign indicating the casino is favored.

The second case might arise in a game with a progressive jackpot. As a simplified hypothetical situation, imagine a $1 side bet with one winning outcome. Say the chance of that result is one out of 100 and the payoff is up to $104. The probabilities are a scant 1/100 of winning versus a strong 99/100 of losing. But the edge is [(1/100)x$104 - (99/100)x$1]/$1 or +5 percent, the plus sign indicating the player is in the catbird seat.

Edge can dog casino aficionados all the way from the machines or tables to the cash kiosks in the lobby. But probability and payoff ratio are the tails that wag the dog. This, because minor shifts in either parameter can yield major changes in edge.

Picture a bet on one spot at roulette. The payoff ratio is 35-to-1. At a single-zero table, the wheel has 37 positions so the chance of winning is 1/37 or 2.70 percent. At a double-zero table, the wheel has 38 positions so the chance of winning is 1/38 or 2.63 percent. The 0.07 percent difference seems trivial. But look how it affects the edge. In a single-zero game, edge is [(1/37)x$35 - (36/37)x$1]/$1 which equals -1/37 or -2.70 percent. In a double-zero game, it's [(1/38)x$35 - (37/38)x$1]/$1 which equals -2/38 or -5.26 percent. The house accordingly averages $2.70 per $100 bet at single-zero tables and nearly twice as much, $5.26 per $100, at their double-zero counterparts.

This effect doesn't depend on a high payoff ratio. Two-column roulette bets, as noted, pay only 0.5-to-1. The probabilities are 24/37 or 64.86 percent and 24/38 or 63.16 percent for single- and double-zero wheels, respectively. Who'd even notice the frequency difference? But the edges, as before, are 2.70 and 5.26 percent almost twice for the double-zero than the single-zero version.

A small change in payoff ratio with a given probability can also have a large impact on edge. Jacks-or-better video poker offers a classic example. Exceptionally savvy players look for games that return 9-for-1 on full houses and 6-for-1 on flushes. With otherwise standard pay tables, the casinos have -0.46 percent edge over players who follow expert or optimum strategy. This means players have average losses of only $0.46 per $100 bet.

When video poker buffs settle for or happen upon 9-5 games, they forego a unit on winning flushes and receive 5-for-1 rather than 6-for-1. No big deal? This little change nearly quadruples the inherent edge to 1.56 percent. With 8-6 pay tables, it's -1.61 percent. And 8-5 schedules give the house 2.71 percent $2.71 per $100 roughly six times more than at 9-6 machines.

Hit rate, the chance of any return whatever, is 45.46 percent in all these variants. The sole difference is a small payoff change for one or two winning hands, figures staring every player in the face but that few bother checking before punching the buttons.

Of course, the probability of winning a particular bet is one thing. That of ending a session or casino visit with more to show for it than a comp for the all-you-can-eat buffet is something else. The dominating factors here are volatility in the short run and edge in the long. Proving that the poet, Sumner A Ingmark, perceived whereof he penned in this provocative punting proverb:


Understand that each bet is just part of the score,

Or risk winning some battles but losing the war.

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 focused on gambling probability and statistics. He passed away in October, 2013.
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 focused on gambling probability and statistics. He passed away in October, 2013.