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Why are 12s Against 2- and 3-up at Blackjack so Perplexing?

11 April 2011

Proficient blackjack buffs sometimes knowingly and rationally violate Basic Strategy. This might occur when the “correct” play is to double down or split a pair but the bettor doesn’t want or can’t afford to risk the extra money for these actions. Another involves prescribed doubles where a player might wish to draw an additional card were it allowed, for instance after receiving a deuce on a 4-5 versus a 6-up; hitting wins more often in these situations although average profit is greater by doubling, so this decision might appeal to a person who prefers or needs a higher probability of a lower gain than the converse.

Whether or not you consider these exceptions legitimate, at least there are defensible rationales for them. If not as a general playing strategy, then at least under the conditions prevailing at a particular point in a game. However, some common violations of Basic Strategy don’t fit these conditions. Prominent among these is standing on a hard 12 versus a dealer’s 2- or 3-up. Basic Strategy calls for hits under these circumstances. Yet experienced players often stand on them.

The underlying reason this occurs is that most solid citizens. They memorize the rules of Basic Strategy but take them as articles of faith. They don’t know about the underlying numbers – that they exist, where they came from, or their values. A popular belief is that teams of experts watched lots of games, noted which decisions worked best for the various player-hand dealer-upcard combinations, and voted when not everybody agreed on the optimum options. It’s also thought, albeit less commonly, that the bosses, who’d rather have players lose than win the close calls, made up Basic Strategy to give themselves an edge. (Otherwise, why would they let players consult their wallet-sized cards during the course of the action?)

In fact, Basic Strategy was derived using computers to run through all possible ways every combination can be resolved with each alternate execution. The program finds the corresponding frequencies of winning, losing, and pushing by comparing the final values of the players’ and dealers’ hands. It multiplies these frequencies by the corresponding payoffs and losses, and adds all the products to obtain the associated “expectation.” This statistical or theoretical figure is generally expressed as a percentage, but can be pictured more intuitively as the average amount won or lost per dollar (or per hundred dollars) bet when the hand is played huge numbers of times. The choice offering the greatest average gain or the least average loss becomes the rule. For uniformity in the cases of doubling and splitting, expectation is calculated with respect to the amount bet at the start of the round, not the total up for grabs at the moment of truth.

For hard 12 (other than 6-6) and 13 against a dealer’s two-up, players could stand, hit, or double. Average losses per $100 bet when the 12 is formed as 10-2 are shown in the accompanying table.

Average loss per $10 bet for 10-2 versus two- and three-up in six- and eight-deck games Figures for 9-3, 8-4, and 7-5 are similar

                                  stand      hit     double
six decks       T,2 vs 2         $29.58   $25.19     $50.37
eight decks     T,2 vs 2         $29.51   $25.22     $50.45
six decks       T,2 vs 3         $25.45   $23.14     $46.27
eight decks     T,2 vs 3         $25.40   $23.19     $46.39

The average loss if the hand is doubled far exceeds that of the other choices; while there may be times when a player might want to double anyway – for instance in the make-or-break round of a tournament – this wouldn’t normally be a conceivable option. Rather, bettors worry about standing or hitting. The data show that both options will lose over the long term. But the penalty is lower by over $4 less per $100 wagered – 4 percent – for hard 12s and $2 for 13s by hitting


So, why do so many bettors stand on these hands? Saving over 4 percent is nothing to scoff at. Many patent decisions in blackjack hinge on offsets this small or smaller. Hardly anyone hesitates to hit rather than stand on 15 against 10-up, but the disparity is only about 3.5 percent.

The primary reason is that players greatly overestimate the likelihood that a dealer’s two- or three-up will bust. In actuality, dealers’ two and three-up have only about 35.2 and 37.3 percent probability of going over 21, respectively. Part of this is because of a general misperception about just how much of an underdog an upcard of six or below makes a dealer. There’s also the notion that the hole card is likely to be a 10, so why not stay in the game by standing and let the dealer take the risk? A third factor is that it’s easy to remember standing on 12 and winning or hitting on this total and losing, showing what’s true anyway – that following basic strategy is no guarantee of success in any specific instance. The poet laureate of the casino scene, Sumner A Ingmark, cleverly captured the core of these arguments like this :

The path of self-deception,
Is paved with vague perception.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.