Dealers' Prospects Mean More in Blackjack than You May Think - Tuesday 7th of February 2006

The tenets of Basic Strategy for Blackjack don't always seem to make sense. Many bettors question whether "the book" is really right about what they're supposed to do in certain situations.

This state of affairs arises because players tend to think mainly about how the alternatives may affect their own totals. Solid citizens generally understand that decisions also depend on dealer upcards. However, few players fully appreciate the significance of the dealer's position. Even experienced blackjack buffs look at the upcard and think exclusively in terms of "stiffs" or "pat hands" twos through sixes or sevens through aces, respectively. They figure that dealers will normally bust with the former and finish in the running with the latter.

Such a simplified model, along with the popular presumption that "the unseen card is a 10," works for lots of conditions. But it fails under just the circumstances players find most perplexing.

As an example, pretend you have a soft 18, ace-seven. This is a tough hand. Depending on the dealer's up-card, Basic Strategy may be to stand, hit, or double. And, you're not alone if you wonder whether there could be a misprint on your wallet-sized chart.

Say the dealer has a nine-up. The chances of a nine yielding the various final totals are as indicated in the accompanying table.

Probabilities of Dealer Final Positions
for representative upcards

final
total
nine-up
six-up
17
12.1%
16.6%
18
11.5%
10.6%
19
35.2%
10.6%
20
12.1
10.2%
21
6.1%
9.7%
bust
23.0%
42.3%

If you stand, the dealer has 53.4 percent chance of beating your 18 with a 19, 20, or 21. Your chance of winning, with the dealer finishing at 17 or busting, is 35.1 percent. So your statistical expectation is to lose 53.4 - 35.1 = 18.3 percent of the time, or 18.3 cents per dollar bet at the start of the round.

Hitting, you have more ways to get weaker than stronger. But the gain in the chance of winning when your hand does improve more than compensates for the greater likelihood it will worsen.

By merely advancing to 19, you win against 17, 18, or bust 46.6 percent and lose against only 20 and 21 18.2 percent. Now you expect to win 46.6 - 18.2 = 28.4 percent of the time, or 28.4 cents per dollar bet. The key is in the high incidence rate of dealer 19s. This dealer total becomes moot when you end with 19. And it's on your side when you finish at 20 or 21. All told, combining your chances of improving and the amount by which you do, you remain an underdog if you hit. But your expected loss is slashed nearly in half, to 9.8 cents per dollar of initial bet.

Consider the dealer's prospects starting with an exposed six. The chances of various final totals are also shown in the table. Standing gives you a win against 17 or bust 58.9 percent, and a loss against 19, 20, or 21 30.5 percent. Your expectation is to win 58.9 - 30.5 = 28.4 percent or 28.4 cents per dollar bet.

You're still projected to win if you hit, but the larger likelihood you'll suffer rather than succeed overshadows the benefit of a higher total beating the dealer more often. This, owing to the absence of a frequency spike like that at 19 with a dealer's nine-up. In all, your expected profit hitting is only 19.1 cents per dollar. Were your choice only to stand or hit, the former would accordingly be preferable at 28.4 cents.

What about doubling? You win less often than by standing; in fact, the probabilities are the same as for hitting. And you must risk twice as much money. However, you're favored when you make the extra bet, and expect to average twice as much profit as if you hit 38.2 cents per dollar. So Basic Strategy is to double since it offers the best expected earnings. The coupleteer, Sumner A Ingmark, contemplated the quandary when he commented:

True gamblers are cautious but not filled with dread,
Of wagering more to get further ahead.

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