Why fewer decks at Blackjack are better, and when they're not

Most blackjack buffs have heard that the fewer the decks from which cards are dealt, the better the shot for players. So, all else being equal, single decks offer the best possible games.

It's true. Some people think only card counters benefit. Those who go this extra step do profit more than others. However, all players get a reduced edge, even folks who flout Basic Strategy. This, owing to changes in the proportions of cards in shoes of various sizes as play progresses, and the resulting effects on blackjacks, doubles, splits, and dealers' chances of busting.

As an example, fewer decks offer greater prospects of being dealt a blackjack. But solid citizens often wonder why it's easier to get blackjacks in decks with four aces than in shoes with 32. And also, since dealers are just as apt to get blackjacks as players, why the impact isn't a wash. The resolution of both puzzles is buried in the probabilities. To understand the phenomenon, you have to recognize that the chance of a blackjack is that of drawing an ace then a 10 plus that of a 10 then an ace.

Here are the figures for a single deck, assuming no knowledge of any cards previously withdrawn from a shoe. Chances are four out of 52 for the initial ace and 16 out of the remaining 51 for the 10. Similarly, chances are 16 out of 52 for the initial 10 and four out of the remaining 51 for the ace. Expressing the values in fractions gives (4/52)x(16/51) + (16/52)x(4/51). Run this on your grade-schooler's Kray supercomputer if you want to verify that it equals 4.82 percent. That's one out of every 20.72 hands.

To get the values for eight decks, note that a pristine shoe this size has 32 aces, 128 10s, and 416 total cards. So the chances are 32 out of 416 for an initial ace and 128 out of the remaining 415 for the complementary 10, plus 128 out of 416 for an initial 10 and 32 out of 415 for the ace. This is (32/416)x(128/415) + (128/416)x(32/415), which is 4.74 percent or once in 21.07 hands.

Now, think about why the edge is reduced by a greater likelihood of blackjacks even though dealers are as liable to get them as players. Dealers with winning blackjacks collect whatever is bet, say $10. Players with winning blackjacks receive half again the amount bet, $15 for $10 wagered. That extra 50 percent is the excess a blackjack is worth to a player relative to a dealer.

Applying the 50 percent to the expected rate at which players expect to receive blackjacks yields the reduction in the edge the house would otherwise have. With one deck, this is half of 4.82 or 2.41 percent. With eight decks, it's half of 4.74 or 2.37 percent. The difference is 0.04 percent.

The nominal edge for a player who adheres to perfect Basic Strategy in eight-deck blackjack is about 0.4 percent. The increased likelihood of blackjacks with a single deck trims 0.04 percent from this amount. It represents a 10 percent discount.

The effects of changes in probabilities resulting from different numbers of decks are even stronger with respect to doubling down. When everything is combined, rules that give the house 0.4 percent advantage with eight decks yield a no-edge game with one deck. Which is why you don't see one-deck games any more.

Except, of course, when the casinos slip in a rule change. For instance, disallowing doubles after splits, requiring dealers to hit soft 17s, or paying only 6-to-5 for winning blackjacks.

The 6-to-5 payoff is especially insidious because it's a 20 instead of a 50 percent bonus for players. Here's the penalty in a one-deck game. A 3-to-2 blackjack shaves edge by (0.5)x(4.74) or 2.37 percent. At 6-to-5, the reduction would be (0.2)x(4.74) or 0.95 percent. Instead of the one-deck having zero edge, players face (2.37 - 0.95) or 1.42 percent. What of the bettors at one-deck 6-to-5 tables who think they know something the bosses would rather obfuscate? They've obviously overlooked this observation from the obstreperous odester, Sumner A Ingmark:

When vital factors are ignored,
A gambler's ox is quickly gored.