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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


Bettors are allowed to split pairs in blackjack and play each split hand separately. This is among the features of the game that moderate the otherwise huge edge that dealers would have by acting last (i.e., after all the players have acted on their hands). But many solid citizens question at least some of the splits dictated by Basic Strategy for four key reasons. Understanding them helps instill confidence, as well as decide whether, or when, it's safe, if not prudent, to flout "the book."


1. Splitting requires risking additional money, because both new hands must have wagers equal to what was bet at the start of the round.

2. Some pairs have totals that already appear strong against the upcard in question.

Betting more to play the halves separately may be past the point of diminishing returns.

3. Some pairs not only have totals that are weak against the upcard in question but the halves, individually, are also underdogs, so the extra bet seems to make matters worse.

4. Proper splitting only cuts the overall edge in the game by about half of one percent so the benefits may not be commensurate with the risk.

The first of these reasons lies outside the realm of mathematical rigor. Proficient blackjack players anticipate augmenting their initial outlays for splits and doubles when it's beneficial to do so. However, instances may arise when this is best avoided; an extreme example might be when losing both sides after splitting a pair would deplete a residual bankroll in one fell swoop.

The second and third reasons follow from the fact that splits fall into three distinct categories:

a) neg-pos  the original total is an underdog when hit or stood upon but the split is projected to be profitable;

b) pos-pos  the original total is favored when you hit or stand but expected earnings increase when the pair is split;

c) neg-neg  the original total and the split are both uphill battles, but the average loss is less in the latter case.

The fourth reason hinges on the difference between joint and conditional probabilities. The overall impact on edge involves the combined effect of each individual split and the chance it will occur. Once a hand is dealt, its incidence is a precondition, and only the gain in expectation by splitting is of interest.

The accompanying table shows all Basic Strategy splits by category, along with the expected gain...

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