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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



Blackjack is a game of fine distinctions. Sure, standing with 10-10 versus 10 is a no-brainer compared to splitting or hitting (both intuitively and statistically). For reference, the math says standing wins 55.80 cents/dollar, while hitting or splitting loses 44.02 and 84.76 cents/dollar, respectively. But many hands are closer calls. Take 9-7 versus 10. It's a dog no matter what you do. With eight decks, your losses average 53.77 cents/dollar standing, and 53.65 cents/dollar hitting. Surrender saves 3.77 cents/dollar over hitting, and hitting saves 0.12 cents/dollar over standing. Some, but not a whole lot of difference, between one and the other.

Six decks have a lower house edge than eight, although the effect is also marginal. With identical rules (this may not be the case; eight-deck games often have weaker options) including resplitting non-aces, the basic strategy house edge is 0.40 percent with six decks and 0.43 percent with eight. Savings are only $0.03/$100 wagered. However, fewer decks mean more to card counters than to basic strategy buffs. The reason is that probabilities change faster and fluctuate more widely with draws from smaller sets of cards, which opens more and better windows of opportunity. Picture it for one or two decks. The chance of getting any rank card is four out of 52 with one fresh deck and eight out of 104 with two. These both equal 7.69 percent. If the first card is a five, what's the chance the next will pair it? With one deck, three out of 51 or 5.88 percent; with two decks, seven out of 103 or 6.80 percent. Fewer decks yield greater probability changes.

Variations in probability with shoe size underlie the other differences as well. For instance, blackjacks pop more often in games with fewer decks. The likelihoods that the player and dealer get a blackjack are equal, but blackjacks impact edge since dealers collect only the amount at risk when their blackjack wins, while players pick up 1.5-to-1 when they luck out and get an untied blackjack.

The chance of a blackjack in a six-deck game, when there are no discards, is known. You can either get an ace then a 10 or a 10 then an ace. That's [(6x4)/(6x52)]x[(6x16)/(6x52 1)] + [(6x16)/(6/52)]x[(6x4)/(6x52 1)] or 4.7489 percent. For eight decks, it's slightly less—substitute 8 for 6 in the above formula and you get 4.7451 percent. The probabilities differ by 0.0038 percent. The edge reduction is half as much, 0.0019 percent ($0.19/$10,000) owing to the half-unit imbalance between what solid citizens and dealers get for their respective blackjacks.

Probabilities of occurrence similarly account for the remaining edge difference between six- and eight-deck games. Blackjack has 550 starting combinations. Of these, six-deck shoes theoretically win more or lose less in 346 whereas eight-deck shoes are projected do better in 190, and the configurations are equal in 14. Of course, combinations aren't uniformly apt to be dealt or yield wins. Six decks are better in 58.1 percent of all hands, eight in 36.5 percent, and the shoes are equal in 5.4 percent.

Proficient blackjack players relish hands on which they can double down. Done properly, these represent auxiliary wagers made when bettors are at an advantage. Basic strategy doubles account for 98 of the 550 combinations, 9.64 percent of all hands, with shoes of both sizes. The benefit in edge arises because 87 of the 98 doubles have higher profit expectation with six decks. The 11 where eight decks are better are 9-2 versus seven, 6-5 versus 10, and 9-2, 8-3, and 7-4 versus eight, nine, or 10.

Occasions to split pairs differ slightly between shoe sizes. Both involve 52 out of the 550 hand combinations. But six-deck splits have a probability of 2.53 percent while it's 2.56 percent with eight decks. For offensive splits, when expectation goes from negative to positive or gets increasingly favorable, six decks account for 37 combinations with a probability of 1.75 percent while for eight decks it's 36 combinations and 1.73 percent. Defensive splits, which reduce but don't reverse the house's advantage, include 15 combinations for six decks at 0.77 percent, and 16 combinations for eight decks at 0.83 percent.

Given a choice, when conditions are equal or superior at six-deck tables, go for these over eight-deck games. But if you must bet more to play, dwell on the definition of "equal" and ponder this pithy proverb from the parsimonious poet, Sumner A Ingmark:

Overbet your pocketbook and run the risk that you'll wish,

You weren't quite as penny-wise while being so pound-foolish.

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