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# Why and how edge at blackjack changes during the course of a shoe

11 June 2012

Probabilities in most casino games don’t fluctuate from round to round. The chance a pair of dice will land with a total of five is four out of 36, regardless of what’s been thrown since the onset of recorded time. Analogously at double-zero roulette, the prospects the ball will come to rest in a red groove on the wheel are 18 out of 38 on every spin. Similarly at the slots, say three reels on a certain machine each have 20 “stops.” The chance of any particular three stops is 1/20 multiplied by itself three times, or one out of 8,000. Make believe that on this machine, Clowns appear at five stops on reel 1, four on reel 2, and one on reel 3. The likelihood of a row of clowns is (5/20)x(4/20)x(1/20) or 20 out of 8,000 – which equals one out of 400. In some cases like these, the nature of the game or the way it’s played may affect your prospects during a round. However, the chances are always the same when you commit yourself to the gamble by making your bet.

Games involving “withdrawal without replacement” are different. Most notably in blackjack, baccarat, and their derivatives, cards are dealt from a shoe during a round, then set aside. As a result they’re unavailable for subsequent rounds, at least until they’re gathered and shuffled to form a new shoe. The composition of the source from which the cards are drawn, and the consequent probabilities, accordingly change from round to round. The effect is especially noticeable – and exploitable – in blackjack because of a correlation between the probabilities that cards of various ranks will be dealt and bettors’ outlook for upcoming hands. In particular, an excess of high values tend to favor players and of low values the house.
The house’s edge arises in blackjack because dealers act last and win when they and the players both bust. In an otherwise balanced game, this edge would be about 5.5 percent. The actual edge against strict basic strategy players at eight-deck tables with fairly liberal rules is only 0.45 percent. The reduction is caused by two factors. 1) Players have options that dealers don’t, including choice of standing on low totals and doubling down or splitting pairs. 2) Uncontested blackjacks pay players 1.5-to-1 but only get the house the amount of the wager; the extra half-bet payoff shifts the edge toward the bettors by half the probability of a blackjack being dealt.

The nominal 0.45 percent edge cited applies to a “neutral shoe” having the original proportions of ranks. It’s valid “off the top” of a shoe, and also as an average over large numbers of shoes from which cards are withdrawn and not replaced during the course of the action. Because 0.45 percent is relatively low, small changes in edge that accompany shifts in probability as a game progresses can shift the advantage to the player. Solid citizens who count the cards withdrawn and no longer available, and therefore know when what remains to be dealt rich in high ranks raise their bets to exploit rounds in which they have an advantage. Conversely when the residual shoe is neutral or rich in twos through sixes, they lower their bets or leave the game.

The influence of shoe composition changes on edge are most readily demonstrated in terms of the benefit afforded by the 1.5-to-1 blackjack payoff. In a neutral shoe, the probability of a blackjack is 4.75 percent. This reduces what would otherwise be the edge in the game by 4.75/2 or 2.38 percent. Assume you’re about to start a round with a freshly shuffled shoe. The distribution of ranks is neutral except that the dealer “burns” the first card. In most casinos, the dealer will show you this card if you ask. Assuming it’s other than an ace or 10, the probability of being dealt a blackjack in that first round rises to 4.77 percent, shaving the house advantage by (4.77 - 4.75)/2 or 0.01 percent. If an ace happens to be burned, the probability of a blackjack falls to 4.62 percent, raising the advantage for the house by (4.75 - 4.62)/2 or 0.065 percent. Alternately, when the burn card is a 10, the probability of a blackjack on the first round is 4.73 percent so the house advantage goes up by (4.75 -4.73)/2 or 0.01 percent. These changes are admittedly small. But we’re only talking about taking one card out of contention.

Instead of just one card being gone, pretend that 52 have been withdrawn. Were the 52 to comprise the statistically correct proportion of 16 10s, four aces, and 32 other ranks, the probability of a blackjack would remain at 4.75 percent and the house’s edge at 0.45 percent. Were the 52 withdrawn cards to include 20 10s and six aces, the probability of a blackjack drops to 4.25 percent. House advantage then jumps up by (4.75 - 4.25)/2 or 0.25 percent, bringing the edge against players up to 0.7 percent. Conversely, if the 52 withdrawals include six 10s, one ace, and 45 other ranks, the probability of a blackjack would be 5.72 percent, reducing edge by (5.72 - 4.75)/2 or 0.97 percent, giving the player an advantage of 0.97 - 0.45 or 0.52 percent. If you’ve been thinking the 1.5-to-1 blackjack payoff is a nice little bonus but no great shakes in the scheme of things, think again. It’s effect on edge is large – and can shift the balance as a shoe gets depleted. Something the poet, Sumner A Ingmark, urged punters to ponder when he penned:
Phenomena dismissed as worthless,
May hide a value ‘neath the surface.