Why Hitting 12 vs Two Beats Standing in Blackjack

No blackjack decision more effectively winnows the wheat among players from the chaff as 12 versus two-up. If this is the way your cards fall in a particular round, you've got a bum rap no matter how you handle it. But some evils are lesser than others.

Basic strategy says to hit. Many solid citizens stand. And card counters know both are correct under appropriate circumstances: stand if the "true count" has risen to the relatively rarified reaches of +3 or above, when the shoe is especially rich in high cards, otherwise hit.

To see why one selection surpasses the other, and by how much, forget card counting or assume the shoe is neutral. I'll explain what the laws of probability governing the entire known universe, including casinos, say to expect in 1001 instances of this hand.

The following list shows the statistically-correct possible final results for a dealer starting with two-up.

final hand	instances
17	140
18	134
19	130
20	124
21	119
bust	354

Here are expected player final hands with single hits on 12.

final hand	instances
13 - 16	308
17	77
18	77
19	77
20	77
21	77
bust	308

You don't have to be a Nikolai Ivanovich Lobachevskii to understand what happens to those who stand and wait. They'll only win if the dealer is kind enough to bust. Expectation is 354 joys minus what's left - 647 sorrows, a net loss of 293 bets.

Hits make life a tad trickier. Players can push, win, or lose.
Push: player and dealer have equal totals of 17 through 21.
Win: player doesn't bust and dealer busts or finishes lower.
Lose: player busts; player reaches a final hand of 20 or below and dealer finishes at a higher non-bust total.

Here's how math mavens calculate instances of pushes. Players expect to finish at 17 in 77 out of 1001 hands. Dealers should end at 17 in 140 out of 1001 hands. The statistically-correct number of pushes on 17 is therefore 77 x (140/1001) or almost 11 out of 1001 hands. Repeating this arithmetic for final scores of 18 through 21, and combining, yields nearly 50 out of 1001 hands.

Here's how numero-noodniks find instances of wins. Players expect to finish at 13 though 17 in 385 out of 1001 hands. In such cases, they win only on dealer busts - 354 out of 1001 hands. Together, these conditions yield 385 x (354/1001) or about 136 out of 1001 hands. Likewise, players expect to finish at 18 in 77 out of 1001 hands. They'll win if the dealer has 17 or busts - 494 out of 1001 hands. Combining yields 77 x (494/1001) or 38 hands. Doing similar calculations for final scores of 18 through 21, and combining, yields 348 winners.

Losses can be figured with a shortcut. They're hands which don't push or win, so the expected total is 1001 - 50 - 348 or 603.

Overall, the expectation for hitting is 348 yippies minus 603 darnits or a net loss of 255. For standing, the net loss was 293. Both are likely losers - but hitting is less onerous by 293 - 255 or 38 out of 1001 decisions. What's this in money? On a $10 bet, hitting offers a theoretical savings of $0.38 over standing.

One more thing. For blackjack geniuses who think this is all simple stuff and want more grist for their mental mills. The various ways to form a two-card non-paired 12 differ in potential to do damage when they're hit against two-up. Ranked from most to least painful, they're 9-3, 8-4, 7-5, and 10-2. Can you figure out why? While you're meditating on this mystery, mull over these mutterings of the immortal muse, Sumner A Ingmark:

Distinguishing 'mongst shades of gray,
Can chase those losing blues away,
And help the canny save the day.