12 vs 2 Blackjack

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One of the most painful hands in blackjack is when you get a 12 versus a dealer who is showing a two?  Oh sure, there are plenty of hands out there that are worse mathematically, since by the book a 12 against a 2 balances out to a loss of just over $0.25 per dollar bet. Lots of player-dealer combinations are far more taxing. For instance, a 16 versus nine, 10, or ace is projected to cost above $0.50 on the dollar. And a 17, which many bettors erroneously think is a safe plateau, bears theoretical penalties for every dollar at risk exceeding $0.38 against eight, roughly $0.42 against a nine or 10, and almost $0.48 against an ace.

The fear of 12 versus two arises from Basic Strategy gently calling for players to hit, while instinct is loudly telling you to stand. The latter, mainly because the possibility of drawing a 10 and busting immediately looms large at four chances out of 13. And, more, prospects for receiving an ace, deuce, three, or four — surviving the peril of busting yet not improving — are similarly strong at four out of 13. That leaves only five ways out of 13 to finish better than where the hand began, although by no means necessarily being a winner.

Intuition about standing is reinforced by the fact that the probability of winning this hand by doing so is actually a bit greater than by hitting. The apparent discrepancy between the likelihood of winning and the expected loss per dollar is resolved when pushes are considered. Hitting may lead to a push, while standing cannot. The accompanying table gives the figures.

Outcome probabilities for 12 versus 2-up

Hit – Stand
chance of winning 34.8% – 35.4%
chance of pushing 5.0% – 0.0%
chance of losing 60.2% – 64.6%
expectation

win minus loss

-25.4% -29.2%

The data show that solid citizens can anticipate winning 35.4 percent of their hands by standing rather than 34.8 percent by hitting. Slightly fewer wins are accounted for primarily by hands that bust when players draw 10s, offset by the impact of those that finish with higher totals than the dealer. What’s left is the effect of the 5 percent of hands predicted to push when players hit, most of which reduces the overall chance of losing. And expectation is the net, on the average, of how much bettors win minus the amount they lose, per dollar put at risk.

Here’s another item of which players interested in the nuances of blackjack might want to be aware. A two-card non-pair 12 can be formed in four unique ways, and expectations differ among them. This results from the shoe starting with a set number of cards of each rank, and those already withdrawn therefore being unavailable for the player or the dealer during the round.

The order for hitting, from least to most desirable along with the statistical loss per dollar bet in an eight-deck game, is 7-5 ($0.253), 8-4 ($0.254), 9-3 ($0.255), and 10-2 ($0.252). Qualitatively, considering only the players’ side, sevens and fives both improve 12s, such that making these less available in the shoe can hurt; at the other extreme, 10s are poison and twos are useless with 12s, so withdrawing these from the cards remaining to be dealt is a benefit.

The order for standing, again from least to most desirable, is the opposite. It’s 10-2 ($0.295), 9-3 ($0.292), 8-4 ($0.291), and 7-5 ($0.289). In this situation, a bettor would like to deprive the dealer of the cards most apt to lead to a pat hand.

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