You'll See More Aces Playing Blackjack with Fewer Decks

16 August 2004

By Alan Krigman

Aces are a big deal in blackjack. They help on both sides of the table by reducing the chance of busting on a draw. In general, though, they benefit the bettors more than the bosses. The ace-10 combination is particularly important because it pays 1.5-to-1 when players receive it, but costs only what's bet at the start of the round when it appears in the dealer's hand. Players also get the opportunity to make favorable soft doubles on some hands containing aces against appropriate dealer upcards. And splitting aces, even with the restriction to one additional card, is projected to earn solid citizens a profit ranging from $0.12 or $0.18 per dollar against an ace or 10 respectively, to $0.68 on the dollar when the dealer is showing a six.

Concern about numbers of available aces causes some skeptics to doubt the dogma decreed by the doyens that downsizing shoes enhances players' chances. For instance, six decks supposedly give the house lower edge than eight decks. And games dealt from single decks offer the lowest edge of all. That is, if you can find casinos with single-deck tables, not spoiled by offsetting rules like 6-to-5 payoffs for blackjacks or no doubles after splits. But some folks wonder whether fewer decks are really better, since they have fewer aces.

They are better; all else being equal, less in this case is more. One reason lies in prospects for a blackjack with different numbers of decks in play. While chances of players and dealers having naturals are the same, greater probabilities of blackjacks favor bettors because of the 3-to-2 payoffs. Here's the skinny for a single deck, assuming no knowledge of cards already withdrawn. The chance of an ace as a first card is 4/52 and that of a 10-value as a second is 16/51; the likelihood of a 10 first is 16/52 and of an ace second is 4/51. Overall, this gives the probability of a blackjack as (4/52)x(16/51)+(16/52)x(4/51) or 4.826 percent. Extending this methodology to six and eight decks, the probabilities are 4.759 and 4.745 percent, respectively. You can see that chances drop slightly as numbers of decks rise.

Another reason involving aces that fewer decks are a boon to bettors follows from the phenomenon of "withdrawal without replacement." Say you play roulette and the ball lands on nine. The nine isn't somehow taken off the wheel for the next spin. But if an ace is dealt in a round of blackjack, it's discarded and isn't available until after the shuffle. So, in the succeeding round, absent a shuffle, one fewer ace remains to be drawn.

Inquiring minds may then want to know the chances, for alternate numbers of decks from which cards are removed and not replaced, that the initial deal on a round will include at least one ace. This, of course, depends on how many spots are in action -- since the number of cards exposed when a round begins will equal twice the spots being played plus two for the dealer. Probabilities of at least one ace for one to seven spots, in games of various numbers of decks, are shown in the accompanying table. Higher probabilities, which favor players, occur with fewer decks.

Probabilities of at least one ace in the initial deal of a round

spots in action
decks	1	2	3	4	5	6	7
1	28.13%	39.72%	49.86%	58.66%	66.24%	72.73%	78.24%
2	27.75%	38.90%	48.52%	56.78%	63.86%	69.91%	75.05%
4	27.57%	38.52%	47.89%	55.91%	62.77%	68.61%	73.59%
6	27.51%	38.39%	47.69%	55.63%	62.42%	68.20%	73.12%
8	27.49%	38.32%	47.59%	55.50%	62.24%	67.99%	72.89%

The effect of shoe size is even more pronounced when the question of interest is the expected or average number of cards between occurrences of aces. The upper limit is 13, which holds for any number of decks with immediate replacement. Without replacement, the figure varies from 10.6 cards between aces for a single deck, to 12.5 and 12.6 for six and eight decks, respectively. Again, fewer decks favor players, with less cards expected between aces.

Naturally, some cynics will still insist that "the bosses wrote the book" and I'm just making this all up because nobody's gonna call my bluff about the math. The best response to them may well be the memorable meter by the immortal muse, Sumner A Ingmark:

The laws of mathematics, when properly applied,
Lead gamblers to conclusions, they'd best not cast aside,
For when they've money riding, facts ought not be denied.