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Hit or Stand: The Hard 16 Dilemma in Blackjack

11 November 2003

By Alan Krigman

Blackjack buffs really hate a hard 16 against a dealer's seven- through ace-up. This, whether the 16 is a 9-7 or 10-6 two-card combination, or a larger set such as 7-8-A or 5-3-4-4. And the antipathy is well warranted. These are the weakest positions in which players can find themselves. They're especially perplexing because everyone knows "the book says hit," yet doing so seems suicidal given that the odds of busting are an adverse 8-to-5. And even surviving a draw doesn't guarantee winning or pushing unless the five happens to appear. Besides, who wrote the book anyway, how do they know, and what's to say the casino bosses didn't pay them to put in things that make players lose?

So, 16 against a high upcard being between a rock and a hard place, just how bad is it to stand? Further, assuming you're tempted to stand, are you better off doing it with some dealer upcards than others? And, if so, which are which?

The answers are in the numbers. After all, besides luck, that's what gambling is about. The numbers for the two-card 16s are shown in the accompanying table. Entries give expected loss per dollar bet when standing or hitting, and the penalty for the latter, against each upcard. Data are for eight-deck shoes with "playable" 10- or ace-up. Playable means the dealer doesn't have a blackjack, since the decision would be then irrelevant. For the cynically inclined, these figures are not made up based on what certain oracles think, or what the bosses say, or what some guru concludes after watching lots of games and jotting down results or tabulating millions of simulated virtual hands; they're found by a "combinatorial analysis" using a computer to increment through and count each and every possible outcome.

Expected losses and penalties per dollar bet, for standing
on or hitting hard 16 versus dealer seven- through ace-up

 
7-up
8-up
9-up
10-up
A-up
stand 9-7
$0.480
$0.513
$0.543
$0.537
$0.663
hit 9-7
0.408
0.454
0.505
0.535
0.514
penalty 9-7
0.072
0.059
0.038
0.002
0.149
stand 9-7
0.476
0.513
0.542
0.541
0.665
hit 9-7
0.409
0.453
0.505
0.535
0.516
penalty 9-7
0.067
0.060
0.037
0.006
0.149

The data are revealing in several respects. On a practical level, they show that the penalty for standing is most severe when the dealer has ace-up, nearly $0.15 on the dollar for either two-card hard 16. Next worst, less than half as bad, occurs against seven. From there, it drops with rising up-cards to under a cent on the dollar, nearly nothing, with a dealer 10. So, if you're to stand at all, do it against a 10 where the mathematical misery is the least; this will happen on about half of your 16s because the shoe has as many 10s as it does sevens, eights, nines, and aces together. And, for reasons that the numbers should make obvious, don't even think about standing on that 16 against an ace.

The table also contains items of interest to solid citizens who want insight into the game and not simply a cookbook of recipes when to do what. For instance, the losses on standing increase as the upcard goes from seven through nine, then fall slightly at 10 and rebound sharply on ace. This effect reflects the chance of the dealer completing a hand without busting, best with ace then nine, 10, eight, and seven in that order.

Losses on hitting also increase as upcards go from seven to eight, nine, ace and 10. This phenomenon tracks the impact of the levels at which dealers are apt to finish when they don't bust, lowest when showing a seven and highest a 10. As completed dealer totals get higher, they become more difficult to beat.

The theoretical costs of flouting basic strategy don't follow either the chance of the dealer busting or finishing at a high level, though. They depend on the differences between the two. Other than against an ace, the penalties fall as the actual expected losses rise. Trade-offs like these make blackjack tough to second-guess and show the danger of applying qualitative logic to quantitative problems. They also explain why so many players wonder whether Basic Strategy really makes sense. The poet, Sumner A Ingmark, envisioned the essence of the enigma like this:

Most systems for gambling are artful delusions,
Applying the numbers yields different conclusions.

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 were focused on those interested in gambling probability and statistics. He passed away in October, 2013.