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# How does fighting or having an edge affect your chances?

12 July 2010

Casinos reap their rewards on the edge in the games they offer. If they know, or estimate, the totals bet over some time period, they can anticipate how much profit will drop to the bottom line.

Predictions based purely on edge are accurate provided three conditions are met. 1) The bosses must get the gross wager right. 2) Small numbers of extremely high rollers don't experience wins or losses so big that they bias the overall results. 3) The selected accounting period is sufficiently long that enough bets are resolved for the law of averages to be legitimately applied.

Solid citizens normally don't gamble enough, especially at the tables, for the law of averages to take hold. In most cases, the issue for bettors is whether they're doomed to lose because of the casino's edge. Less often, for advantage players such as card counters at blackjack, the question is if they're guaranteed to succeed because of their edge over the house. The answers in both instances involve the relationship between edge and volatility.

Pretend you follow perfect Basic Strategy at blackjack, starting each coup with a \$25 bet. You fight about 0.5 percent edge. In a hypothetical statistically-correct game, after some number of bets (call the number "n"), edge will have eroded your fortune by 0.005 x \$25 x n. But, owing to volatility, you may be far above or below this level. The effect can be quantified using a metric known as "standard deviation." For n blackjack coups at \$25 each, standard deviation equals 1.13 x \$25 x (the square root of n).

According to the "empirical rule" of statistics, the chance of being within one standard deviation above or below the average is roughly 68 percent. Half of the remaining 34 percent -- that is, 17 percent -- will be more than one standard deviation over the average; an equal fraction will be a greater amount below it.

This information, and a little algebra, give the number of rounds for which an upswing due to volatility will have less than 17 percent prospects of overcoming the loss due to edge. Set 0.005 x \$25 x n equal to 1.13 x \$25 x (the square root of n) and solve for n. The answer is 51,076 rounds. Fewer hands give you over 17 percent chance of being ahead; more coups dim your outlook.

Suppose, instead, you get 1 percent advantage over the house by counting cards. To do this, you have to bet according to the promise of the ranks left to be drawn. You size your wagers on a sliding scale, lowest when the deck is most adverse, highest when you're most strongly favored. Assume, arguendo, you get 1 percent edge with a bet spread of \$10 -- 28 percent, \$15 -- 28 percent, \$25 -- 26 percent, \$50 -- 13 percent, and \$100 -- 5 percent.

Your average wager with this distribution, by no didactic coincidence. is \$25 per round. Edge would put you 0.01 x \$25 x n ahead after n rounds. The standard deviation of the distribution, itself, is 1.31 x \$25; that of n hands is 1.13 x 1.31 x \$25 x (the square root of n). Now -- that darned algebra again -- equate gain from edge to the loss corresponding to one standard deviation and solve for n. The result, 22,055, is the number of rounds you have to play before you have less than 17 percent probability that a downswing due to volatility puts you in the hole. Play fewer hands and the possibility of a loss is larger; undergo more hands and you're less liable to finish behind.

These figures may help explain why you can prevail despite the casino's edge and fail when it's yours. Perhaps you think this says card counting isn't worth the bother. Maybe for you, it isn't. But consider that the more you play, the less you're apt to win as an underdog and lose when you're sitting pretty. Further, the values cited only tell part of the story. You have an even chance of being below or above the level predicted by edge. So, no matter how long you play, you've a 50 percent shot at being further down or up than edge suggests. Forfeiting less or earning more, or... well, you know. All of which motivated the multifaceted muse, Sumner A Ingmark, thusly to advocate evaluating alternate outcomes in tuning your tactics:
Strategies that prove effective.
Oft have more than one objective.