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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


An oddity of blackjack is that cards may be favorable in some situations, and adverse in others. Card counters know that shoes rich at the high end help them, while those with excess of low card-ranks serve the house, so they raise or lower their bets accordingly.

But, consider this. Say you're dealt a 5-6 and double-down. You'd likely to pull a 10. However, you need those fives and sixes to get into this catbird seat. And, if you pulled 6-6 against a three, you'd like to split the pair in which case a 10 on one or both splits wouldn't knock you out, but would certainly leave you vulnerable.

Regardless, solid citizens generally like to see aces coming at them. One reason is that a pair of aces, when split, tends to be profitable. Another is that aces offer a second chance in soft hands; for instance, pulling a five or six to A-4 would be great, but seven or above leaves a new shot at drawing successfully.

The most evident benefit of aces, arguably the most important, is their role in blackjacks. Everyone likes to get blackjacks. The worst they do is push if the dealer also lucks out and gets a blackjack. Otherwise, they win automatically and, most importantly, it pays 3-to-2 a half-bet bonus.

An indirect boon of this bonus is its impact on the house advantage. Even if blackjacks paid even money, they'd be coveted anyway. Bu, they'd have no impact on edge because players and dealers have equal prospects of getting them.

With bettors winning 3-to-2 on their blackjacks, while losing only what's at risk, blackjacks in and of themselves cut the inherent edge in the game by...

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