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Playing It Smart: Why almost everyone questions at least some blackjack pair splits6 July 2009
Bettors are allowed to split pairs in blackjack and play each side separately. This is among the features of the game that moderate the otherwise huge edge dealers would have by acting last. But many solid citizens question at least some of the splits dictated by Basic Strategy. This, for four key reasons. Understanding these 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 hit or stood upon 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. Hands within each group are listed in ascending order of improvement. Note that the gain in expectation from the split is equivalent to the profit projected per dollar of the supplementary bet made to split the pair, so players always have an edge on the secondary wager.
Splits, by category, with theoretical gain in expectation (eight decks, resplitting and doubling after splits allowed)
class, pair (cents gain in expectation per dollar initial bet)
pos-pos 4-4 v 5 (2.4), 4-4 v 6 (4.2), 9-9 v 2 (7.4), 9-9 v 3 (10.8), 9-9 v 8 (13.0), 9-9 v 4 (14.7), 9-9 v 6 (18.5), 9-9 v 5 (19.4), 2-2 v 6 (22.4), A-A v 8 (26.2), A-A v 7 (30.8), A-A v 2 (39.9), A-A v 3 (42.4), A-A v 4 (45.0), A-A v 5 (46.9), A-A v 6 (49.1)
neg-pos 2-2 v 7 (9.7), 2-2 v 4 (11.4), 3-3 v 4 (11.5), A-A v A (14.4), 2-2 v 5 (17.5), 3-3 v 5 (17.7), 3-3 v 6 (22.2), A-A v 9 (23.5), A-A v 10 (24.9), 6-6 v 5 (26.4), 7-7 v 4 (27.2), 6-6 v 6 (31.2), 7-7 v 5 (32.2), 8-8 v 2 (37.1), 8-8 v 3 (40.2), 7-7 v 6 (40.5), 8-8 v 4 (43.5), 8-8 v 5 (47.0), 8-8 v 6 (56.4), 8-8 v 7 (73.1)
neg-neg 3-3 v 2 (0.7), 2-2 v 2 (3.6), 6-6 v 2 (5.3), 3-3 v 3 (5.7), 8-8 v 10 (6.0), 2-2 v 3 (7.4), 3-3 v 7 (10.2), 9-9 v 9 (10.4), 8-8 v 9 (11.8), 6-6 v 3 (12.8), 8-8 v A (14.8), 7-7 v 2 (16.8), 6-6 v 4 (20.3), 7-7 v 3 (21.7), 7-7 v 7 (28.1), 8-8 v 8 (42.8)
Neg-pos splits rarely cause dilemmas. The strongest of these is 8-8 versus 7-up. With $10 bet, expectation goes from losing $4.10 on a hit, to winning $3.21. Neg-neg splits are the toughest calls. The weakest of these is 3-3 versus 2-up. The auxiliary $10 cuts the average loss by $0.07, savings not everybody considers worth the extra cash up for grabs. The dreaded 8-8 versus 10-up is also a neg-neg split. On a $10 bet, theoretical loss is $5.36 bet hitting and $4.76 splitting. Splitting saves a $10 bettor an average of $0.60. Were expectation the only decision criterion, it would be the right move. But, can a player be faulted for not viewing $0.60 on the average worth risking $10 here and now? The eminent epigramist, Sumner A Ingmark, envisioned the enigma thus:
Should you hold decrees immutable,
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