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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

Some blackjack buffs - casino bosses, too - believe card counters can consistently beat the house. Not so. Counting yields a modest edge or advantage or over the casino. But, edge needs numerous decisions to earn money reliably.

In the short term of a single session or casino visit, edge is dwarfed by volatility - the bankroll up or down swings - that occur whenever a wager is resolved. To see this, picture a $100 even-money bet with 50.5 percent chance to win and 49.5 percent to lose. Bettors have a 1 percent edge and theoretically earn $1 per coup. But, they actually either win or lose $100. In a small number of rounds, nobody notices the $1.

That's not all. Solid citizens also have limited bankrolls. The law of large numbers presupposes that they can continue playing until edge overwhelms volatility. But this doesn't hold for those who run out of money during normal cold spells. The following three scenarios show how this works.

  1. Basic Strategy Player. Players are at 0.4 percent disadvantage adhering to perfect Basic Strategy. If they finish 400 rounds betting a flat $20 in each, their theoretical loss due to edge will be 0.4 percent of the $8,000 total amount wagered or $32.
  2. Conservative Card Counter. Conservative card counters might get a 0.5 percent edge by pressing their bets cautiously on rising counts. A reasonable 1-4 betting spread with frequencies with which each bet is made is: $10 (40 percent), $20 (30 percent), $30 (20 percent), and $40 (10 percent). The average wager for this bet spread is roughly $20. If these players complete 400 rounds, their theoretical earnings due to their 0.5 percent edge will be $40 ($8,000 times 0.5 percent).
  3. Aggressive Card Counters. More aggressive card counters might achieve a 1 percent advantage with the following 1-8 betting spread and fractions of hands on which they're made: $10 (55 percent), $20 (25 percent), $40 (9 percent), $50 (6 percent), $60 (4 percent), and $80 (1 percent). The average bet is again about $20. If such players complete 400 rounds, their theoretical earnings due to edge will be 1 percent of $8,000 or $80.

The accompanying table shows the results of a "risk of ruin" analysis for each of the above scenarios...

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