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# How does the “count” in blackjack affect the edge?

13 June 2011

House edge in blackjack arises from the “double bust.” Players finish their hands before dealers. A player who busts therefore loses, even if the dealer subsequently does, too.

When ranks of cards in the supply to be dealt are in their original proportions, drawing to all totals below 17 and standing at or above this level – except for hitting soft 17 – yields 28 percent chance of busting. The probability of a double bust is 28 percent times 28 percent, which equals 7.84 percent. In practice, this figure is reduced by the bonus paid to players for winning blackjacks and by the options solid citizens may exercise, when statistically appropriate, to stand on totals below 17, split pairs, double down, and – in enlightened establishments – surrender.

Other than at the relatively few tables having continuous shuffling machines, the complements of ranks available to be dealt change as cards are withdrawn and not replaced in the supply. The changes affect the probability of busting and of receiving a blackjack, and also impact both the likelihood of situations calling for and the efficacy of the various player options.

As a simple example, consider the prospects for a blackjack a) in a fresh six-deck shoe from which a five has been burned, and b) in one having 19 low ranks plus the burned five, and no aces or 10-values, withdrawn. The probability of a blackjack in case (a) is [(4 x 6)/(52 x 6 - 1)] x [(16 x 6)/(52 x 6 - 2)] + [(16 x 6)/(52 x 6 - 1)] x [(4 x 6)/(52 x 6 -2)], which equals 4.78 percent; blackjacks paying 3-to-2, a 50 percent bonus, trim the edge by 2.36 percent. The probability of a blackjack in case (b) is [(4 x 6)/(52 x 6 - 20)] x [(16 x 6)/(52 x 6 - 21)] + [(16 x 6)/(52 x 6 - 20)] x [(4 x 6)/(52 x 6 - 21)], which equals 5.42 percent; a 50 percent blackjack bonus shaves the edge by 2.71 percent. The excess of 10s and aces relative to cards of lower rank has favored the player by an additional 2.71 - 2.36 or 0.35 percent based solely on the expected frequency of blackjacks.

Overall, a reservoir of cards richer than normal in population of high ranks reduces house advantage; heavy low-ranked content raises it. This is the basis of card counting. In practice, bettors tally the net of high and low cards utilized – typically by starting with zero, adding one when a rank from two through six is withdrawn, and subtracting one when an ace or 10-value becomes unavailable. Relative to a neutral distribution, positive totals indicate increased density of high ranks and corresponding probability of their being dealt; conversely for negative sums.

The primary response to a high count is to bet more, and to a low count to bet less or skip a round entirely. A secondary response is to modify Basic Strategy for conditions when the bias in proportions of ranks substantively alters the probabilities and resulting expectations of outcomes.

Running counts don’t provide enough information to change bet sizes or modify playing strategies. This, for instance, because a running count of six has a weaker impact when four than three decks remain in the shoe. With four decks, the increase in the probability of drawing a 10 or ace is 6/(4 x 52) or 2.88 percent; with three decks, it’s 6/(3 x 52) or 3.85 percent. To account for this phenomenon, card counters divide the running total by the number of decks in the remaining portion of the shoe. The quotient, typically taken to the half-shoe, is the “true count.”
Inquiring minds, of course, want to know how good or bad blackjack gets as the count rises or falls. A good approximation for the change in edge relative to the nominal fresh shoe value is simply half the true count. For a true count of +5, this would be +5/2 or +2.5 percent. If the nominal edge is -0.5 percent, that with the +5 true count would be -0.5 + 2.5 or 2 percent. A true count of -3 would mean the edge worsened by -3/2 or -1.5 percent, and equaled -2 percent.

Some blackjack aficionados aspiring to card counting wonder whether they should simply bet at two levels. A minimum – possibly zero – when the house has the edge and a maximum when they’re in the catbird seat. Given an infinite bankroll, this could work. But for folks with ordinary stakes, the danger would be too great of going broke during the normal downswings induced by volatility, notwithstanding the advantage. It’s therefore more common to vary bets in stages, increasing or decreasing them with the true count. One approach is to establish a base wager and think of it as one unit. Then bet the true count minus one unit at any juncture.

You can go further with more elaborate counting methods and use of “index” levels at which to modify your strategy. Unless you’re copiously funded and play blackjack for a living, these enhancements may be well past the point of diminishing returns. Especially because casinos are paranoid about card counters. You may consequently find shoes being shuffled when conditions get too advantageous for you, or your maximum bet being so limited you can’t make any real dough, anyway. Here’s how that perceptive poet, Sumner A Ingmark, put this perplexment:

When gambling tools at your behest, bring big bucks to your treasure chest,
Panic overwhelms the bosses, who then try to stem their losses.