Stay informed with the
NEW Casino City Times newsletter!
Best of Alan Krigman
You'll See More Aces Playing Blackjack with Fewer Decks16 August 2004
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.
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
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,
Best of Alan Krigman