Fighting the Casino on Two Fronts, Odds and Edge

Casino gambling is usually a two-front fight. With certain exceptions, players must surmount adverse odds while also weathering the erosive effects of house edge. Little about casinos boggles the brains of betting buffs as badly as the distinction between these two factors. Here it is, simply put. Odds indicate the chances of losing versus winning. Edge measures the theoretical commission the house earns by paying winners less than the inverse of the odds they must overcome.

Which is more important? Probability theory suggests that the quality of a bet depends on edge, while odds are irrelevant. Decision theory intimates that odds, coupled with payoffs, dominate and edge hardly matters. In the real world, if casinos can be considered such, each has a place. And, understanding both helps explain why solid citizens willingly subject themselves to the associated adversities. I'll explain, showing how and where they fit, using keno as an example.

First, think about odds. Make believe you bet on a 10-spot keno card. That is, you pick 10 of the 80 possible numbers. The unseen hand of fate, known to the cognoscenti as a computer, selects 20 numbers. The following list shows the approximate odds against your having zero through 10 matches.

You win if you have at least five matches. And the more you have than five, the more money you collect. The odds are roughly 14-to-1 against any win at all. They are as shown for various levels of success -- up to nearly 9 million-to-1 against matching all 10 of your picks. These odds seem discouragingly steep, yet casinos keep printing new keno cards to accommodate the crowds.

The reason rational people play, despite the odds, is that the rewards for multiple matches are munificent. While hitting five out of 10 -- an 18-to-1 shot -- only pays even money, matching all 10 returns $25,000 for every dollar bet. And lots of folks are willing to toss a buck into an abyss if there's even a remote chance $25,000 will come bouncing back. This isn't just a matter of greed, fantasy, or desperation. Decision theory judges this to be a good bet for those who consider the "utility" of $25,000 greater than that of 25,000 times $1. Similarly for consolation prizes like $2,500 for nine matches and $1,300 for eight.

Edge enters the picture when you appraise how much of the punting pool eventually goes back to players. If payoffs were consistent with odds, everything taken in would be pooled then redistributed and edge would be zero. The casino wouldn't earn a cent. Even so, of every million hopefuls who bought 10-spot cards for $1 each, 935,300 would lose. The other 64,700 would split the $1,000,000.

This isn't how it works, though. On 10-number tickets, averaged over a plethora of players, the casino pockets a tad over $0.20 per dollar bet and disburses the rest. Of the million who bought 10-number cards at $1 each, 935,300 would still lose. But now the lucky 64,700 only split $800,000. The casino keeps $200,000. The house edge is the percentage of the total bet the bosses hold back -- in this case 20 percent -- $0.20 per dollar, or $200,000 per million. Those who get $25,000 neither know nor care that were this a fair contest, they might have made $35,000 or more.

Does 20 percent edge seem too high? At blackjack, good basic strategy can trim edge to half of one percent. Likewise for craps bets on pass or don't pass with full odds. But neither game offers an opportunity to drag a sawbuck across the casino floor and attract $250,000 -- not to mention a comp for two at the all-you-can-eat buffet -- in five minutes. It's as the Keats of keno, Sumner A Ingmark, cunningly commented: