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By Alan Krigman

Alan Krigman, and his poetic sidekick Sumner A. Ingmark, have been illuminating the dark recesses of casino gambling for more than a dozen years. Mr. Krigman is especially well known for sharing his insights into the mathematics underlying the various games (including blackjack), the influence of volatility and skewness as well as edge on bankroll during the course of a session, and the impact of betting as well as decision strategies on expected performance. A searchable archive of Mr. Krigman's prose and Mr. Ingmark's muse is online at


Edge is what makes casinos win. But is it what makes players lose? Collectively yes, because gambling is an instance of what economists call a "zero-sum game." For punters individually, the answer is No.

People may indeed lose exactly, or close to, the product of edge times their gross wagers during a session. But, it would be purely coincidental. Rather, a few solid citizens win moderate to large amounts, while the majority lose most, or all, of their stakes. This is a result of the volatility of the action, which swamps edge over the statistically small number of coups folks encounter during particular sessions or casino visits. It averages out for the casino, but not in a way that a single gambler would notice.

Some games and bets have inherently high volatility. Slots and table games with low-probability jackpot payoffs, or long shots, often induce wild bankroll swings. You can win big with a few 35-to-1 single-spot roulette payoffs. Or, waiting for such hits, you can lose far more than 5.26 percent of the total amount that you bet.

In other games, where the usual payoffs are 1-to-1, volatility is inherently low. You can raise volatility yourself for a given average wager by simply varying your bets. To illustrate in blackjack, say you slide out a flat $15 for 100 rounds. Chances are roughly two out of three that your bankroll will be between $163 up and $177 down. If instead you bet 20 rounds each at $5, $10, $15, $20, and $25, your average wager is still $15, but chances are two out of three that your swings will range within $180 up and $194 down.

Doubles and splits are another way to boost blackjack volatility...

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