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Best of Alan Krigman

11 July 2011

The most frequently occurring pairs in blackjack are 10s. The probability of receiving such a pair is 9.40 percent. Against a dealer’s 10, it’s 2.85 percent, against any other rank it’s 0.73 percent individually and 6.55 percent collectively.

The “book” decrees standing on the 20 and not splitting the pair. Most solid citizens obey this edict although some wonder whether splitting the pair might not really be such a bad idea. Especially if the dealer has an upcard other than a 10 or ace, because the likelihood of splitting 10s and winding up with two rather than only one strong hand seems to be reasonably good. Add to this the notion that Basic Strategy is to split nines against anything except seven, 10, or ace, and starting each half with a nine is patently less promising than beginning with a 10.

Say you do split 10s against a dealer’s two through nine. You automatically receive a second card on each half of the split. The chance of pulling a 10 on both sides for two 20s is 9.18 percent. The outlook for a 10 on one and an ace on the other for a 20 and a 21 is 4.74 percent. And the probability you’ll draw an ace on both 10s for two 21s is 0.58 percent. Combined, you’ll stay the same or improve on both new hands with the single mandatory card is 14.51 percent.

Assume the dealer shows seven or over so you take your mandatory card then draw if necessary until you get to at least 17 or you bust. Your chances of reaching the possible end points on either side are 11.2 percent each for 17, 18, 19, or 21; 34.0 percent for 20; and 21.2 percent for a bust.

All told, you may imagine these potentials look fairly decent. But intuition, even when bolstered by some data, isn’t how NASA gets space probes close enough to the moons of Jupiter to send photos of their surfaces back to earth. And, notwithstanding the results of particular lucky rounds of blackjack, it won’t help players optimize their performance in the long run, either.

Examination of the average or statistically expected results of standing or splitting 10s against the range of dealer upcards affords more reliable and definitive guidance about how to play blackjack successfully. The values in the accompanying table provide the desired information, in terms of projected gain or loss per dollar bet at the start of the round.

Expected win or loss per dollar bet, standing on or splitting pairs of 10s

upcard   stand     split
2      + \$0.64   + \$0.07
3      + \$0.65   + \$0.16
4      + \$0.66   + \$0.25
5      + \$0.67   + \$0.34
6      + \$0.70   + \$0.43
7      + \$0.77   + \$0.25
8      + \$0.79   - \$0.01
9      + \$0.76   - \$0.30
10     + \$0.56   - \$0.44
ace    + \$0.65   - \$0.34

The data show where qualitative logic can lead you awry. Against dealer upcards from two through seven, splitting the 10s is indeed favorable and expected to yield a gain, on the average over many hands; the fly in the ointment is that the profitability is less than anticipated by standing. For eight to ace-up, standing on the 20 favors players while splitting is detrimental.

The argument that splitting is recommended for nines against most upcards – so, why not for 10s under comparable conditions – is also flawed. It overlooks three salient factors:

• A total of 18 is nowhere near as strong as a 20 because the former can be beaten by three “made” hands and the latter by only one.

• Starting with nine and drawing to 17 or above has prospects of 11.7 percent of returning to 18; chances of improving are 35.2 percent of finishing at 19, 12.1 percent at 20, and 6.1 percent at 21 – a combined 53.4 percent. Starting with 10 and drawing to 17 or above is less promising; chances are 34.0 percent of returning to 20 and 11.2 percent of improving to 21.

• The mandatory draw to a nine may be a two, for a total of 11 – a candidate for a strong double; nothing drawn to a 10 yields a legitimate opportunity to double down.

The bottom line here is not to split 10s if your criterion is to maximize your expectation. A footnote below the bottom line is that you’re still favored splitting the pair against two through seven, to a small degree when the upcard is a two and moderately when it’s three or above.

So, if you have a compelling incentive to double the money at risk shooting for twice the gain, knowing you may be sacrificing the chances with a strong 20, well ... that’s why they call it gambling. And why the illustrious inkslinger, Sumner A Ingmark, ingenuously intoned:

Though the rule book’s not a joke,
Those impelled to go for broke,