How Much Must You Beat the Odds to Overcome the Edge?

Casino games are structured so the house has an edge. Meaning that the moguls ultimately get the money. Clearheaded folks, among others, rush to the tables or machines anyway. It can't just be for those coveted comps to chow down free at the fabled all-you-can-eat buffets, or for exclusive invitations to queue up and get toaster ovens anybody who really wants one already has.

Not that these things don't matter. But there's also the fact that solid citizens can win by overcoming, without eliminating or reversing, the edge tilting the playing field against them. This can happen because two primary mechanisms work, over series of decisions, in every game of chance. Specifically, edge acts like a small and typically hidden fee the casino charges in return for running the games, and volatility accounts for the fluctuations players' bankrolls undergo as they win and lose individual bets.

Picture a $30 Buy bet on the four at craps. The source of edge, here, is $1 "vigorish" the house charges, up front, to book the bet. Volatility occurs because a win earns you $60 while a loss costs you the $30. In any one round, considering both edge and volatility, each decision swings your bankroll up by $59 or down by $31. The $30 or $60 volatility essentially swamps the $1 edge.

As the number of rounds increases, the gross impact of edge grows at a more rapid rate than that of volatility. The reason is that edge works inexorably in the same direction, while volatility goes both ways. So edge mounts with extended play, but volatility can go to and fro - tending more toward a wash than the extremes of sadly unremitting losses or gloriously steady wins.

When the number of bets gets sufficiently large, edge therefore becomes dominant and decreasingly likely to be surmounted by volatility. But recreational gamblers don't generally experience enough decisions in their casino careers, let alone in single sessions, to be in the range where an upswing due to volatility - albeit of low probability - can't save their hides.

Chance of success or failure during a session or casino visit can be calculated precisely from edge, volatility, and number of decisions. But much can also be said for rough guidelines you can figure for yourself and readily envision, without a degree in rocket science. One such "rule of thumb" is to estimate how many wins are needed over the "statistically correct" number, to overcome the edge in the decisions encountered during a representative time period. This, along with fact or fancy about the frequency with which a bet wins or loses, gives you an idea of your prospects for victory. The approach can be used if you know either how often a wager is "expected" to win in a given number of trials, or have a handle on the value of edge.

The $30 Buy on the four at craps illustrates the first situation. On the average, 144 throws of the dice (around two hours' action) yield 36 decisions, 12 fours and 24 sevens. Players "expect" 12 $60 wins and 24 $30 losses. This would break even on volatility and lose $36 on the $1 vigorish. If 144 throws had 37 decisions, one extra win along with the 24 expected losses, players would finish with $23 profit ($60 from volatility minus $37 from edge). More wins or fewer losses, the desired conditions, are gravy.

Blackjack offers a case where you might know the edge, but not the "expected" frequency of wins. Say you play perfect Basic Strategy with moderately liberal rules. Edge would be 0.5 percent - the house theoretically earns 1/200 of your bet per round. In 200 rounds (about two hours at a table with three other people) the bosses figure you as worth one bet unit. You have to win only one "excess" hand in those 200 rounds to cancel the edge. Play looser and the casino might have 1 percent edge - a theoretical take of one unit every 100 rounds. In 200 hands, you need two supernumerary successes to nullify edge.

This rule of thumb lets you compare alternate games and bets. It also suggests when you've been hot and should quit with a profit, or cold and should cut your losses. It's only a rough guideline. But, as the versifier of the vigorish, Sumner A Ingmark, voiced:

When puzzled by enumeration,
Seek comfort in approximation.