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# How To Estimate the Chance You'll Fall Behind, and Stay There

16 December 2003

Nobody likes leaving a casino with less money than they brought. Especially when they were ahead at some point and didn't quit. But solid citizens may also step off the cliff at the start and never get back. Although they may blame practical handicaps such as the clock running out or the fanny pack running dry, and these may indeed be factors, a more fundamental phenomenon is afoot.

Assume a gambler had unlimited time and resources, and the house had no inherent advantage in the game. It then would be possible, in principle, for the bettor to regain any losses by continuing to play. When the casino has an edge, however, even a bettor with no temporal or monetary restraints faces a finite risk of losing the first round, and never subsequently seeing the light of day.

Of course, nobody can keep going indefinitely. But conservative players can approach this ideal. So, chances of hitting the skids and never recovering in the theoretical case offer reasonable baseline thresholds for anticipating real performance.

If a person with no practical constraints makes only even-money bets, for the same amount on every coup, the chance of falling behind on the first round and never rebounding can be estimated using a simple formula. The calculation is c = 1 - (2 x p), where p is the probability of winning each round. The chance, subject to these restrictions, equals the value of edge in the game.

Bets on Player at baccarat fit the paradigm. They pay 1-to-1 and have 49.42 percent probability of winning. A gambler flat-betting Player therefore has a chance of always being behind of c = 1 - (2 x 0.4942) or 1.16 percent. Similarly for Pass at craps with no odds. Here, the probability of winning each coup is 49.29 percent. Someone making identical line bets on every come-out therefore has a chance of being permanently down of c = 1 - (2 x 0.4929) or 1.42 percent. Outside bets such as red or black at single-zero roulette also pay 1-to-1. The probability of winning any round is 18/37 or 48.65 percent. The chance that a person betting a constant amount on every spin will never be even or ahead is then c = 1 - (2 x 0.4865), or 2.7 percent.

The likelihood of sailing into the wind for a whole session may be more or less than the value of edge for bets other than even money. The basic formula is still in the ballpark. But the wager must be converted to its "even money equivalent." This is the size of a bet paying 1-to-1, and the associated probability of winning, that would have comparable effect on bankroll swings.

Consider a "lay," a bet that pays less than the amount at risk but is projected to have a success rate over 50 percent. An example might be a \$41 "no four" at craps. The odds of winning favor the player by 2-to-1, but a \$41 outlay nets only \$19. Edge is 2.4 percent. The even-money equivalent is \$28.30 with a 48.23 percent probability of bringing home the bacon. The chance a player making this bet exclusively will start with a loss and never see a profit is 3.53 percent -- numerically higher than the 2.4 percent edge. Lay bets pay less than what's up for grabs and accordingly require more wins to compensate for each loss. A result is that edge takes an additional toll on the road home.

Longshots behave oppositely because one or a few hits can get a player back after several losses. For example, bet \$1 on a three-number row at single-zero roulette. The edge is the same 2.7 percent as with the bet on red or black. A win pays \$11 and has an 8.1 percent shot at occurring. The even-money equivalent is \$3.28 with a 49.59 percent probability of winning. The chance a persistent, well-heeled player will always be in the dog house with this bet is 0.82 percent -- far below the 2.7 percent edge.

More time and money can trim the likelihood of starting a game on the downside and staying there, as do bets with inherently lower edge or higher volatility. But, if the casino has an advantage, at least some prospect looms of this fate befalling every gambler. So, as the punters' poet, Sumner A Ingmark, observed:

Sophisticated gamblers know,
The points past which they cannot go.

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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.