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# You Can Learn Some Subtleties of Blackjack from Two-Card 15s

16 August 2006

A starting total of 15 in blackjack is no cause for joy. If it's "hard" -- 10-5, 9-6, or 8-7 -- expectation is negative even though executed optimally. That is, players expect to lose more than they win. Soft 15 -- A-4 -- is a bit better. Expectation is negative against a dealer's two, eight, nine, 10, or ace; however, it's positive by hitting against three through seven, and more so by doubling if the dealer shows a four, five, or six.

Basic Strategy, the ideal set of decisions, is fairly straightforward. For hard 15, stand against a dealer's two through six and hit against seven through ace; in casinos offering late surrender, use this option against 10-up. For soft 15, double against four through six and hit otherwise.

If all you want is a cookbook telling the best way to play blackjack, don't bother reading any further. You've now gotten the rules for your 15s and all that's needed is to memorize them.

But, there's a grab bag (or is it a Pandora's Box?) of insights into the nuances of the game you can gain by exploring this total more deeply. The effects are embodied in the accompanying table. Entries are values of expectation, the percentage of an initial bet players win (plus) or lose (minus), on the average, for each two-card 15 and dealer upcard combination. The data assume eight decks and rounds in which dealers don't have blackjacks.

Expectation (average percent of initial bet players win or lose)
for two-card 15s against various dealer upcards

player hand

 upcard A-4 10-5 9-6 8-7 2 -0.146% -29.306% -29.051% -29.038% 3 +2.840% -25.168% -25.144% -24.888% 4 +6.138% -20.876% -20.855% -20.846% 5 +13.176% -16.364% -16.366% -16.606% 6 +18.211% -15.367% -15.595% -15.606% 7 +3.656% -36.914% -36.900% -36.437% 8 -2.807% -41.688% -41.726% -41.242% 9 -11.236% -47.203% -47.255% -46.819% 10 -17.335% -50.405% -50.512% -50.095% A -13.321% -48.219% -48.187% -47.707%

One fact jumping out of the table is that the dealer upcard matters. A lot. Hard 15s are all underdogs, but the severity of the disadvantage is least against a dealer's six, worsens as the upcard drops to two, jumps and steadily increases as it advances from seven through 10, then falls back slightly on an ace. With soft 15s, properly executed doubles offer reasonable profit potential but this declines as the dealer's upcard drops from six to five to four. Hits against three- and four-up are advantageous for the player, but weakly so. And A-4 is at a disadvantage against two and eight through ace-10 being heaviest.

Another point of interest, which few solid citizens realize, is that the constituents of a two-card 15 make a difference. In some instances more than others. The effect is strongest when the statistically correct play is to hit. Consider, for instance, hard 15 versus eight-up. Per \$1,000 bet, players are projected to lose \$416.88 on 10-5, \$417.26 on 9-6, and \$412.42 on 8-7. Think about why this is true. A 10-5 and 9-6 each remove a desirable card from the shoe -- a five and a six, respectively -- making it less likely that either will appear on a hit. Conversely, the 8-7 takes two undesirable cards out of contention, making it less likely that hitting will cause the player to bust.

Finally, the table shows why surrender is superior to hitting the hard 15 against a 10. Expectation when hitting is to lose more than 50 cents per dollar bet on the average. It's 50.405 cents for 10-5, 50.512 cents for 9-6, and 50.095 cents for 8-7. Surrender costs exactly 50 cents on the dollar, and therefore represents less projected loss than hitting.

Maybe a fraction of a cent out of a dollar seems like chicken feed to big shots who came to gamble and not surrender, or who like to double-down and go for broke when "the book" says just to hit. But those are the folks who flagrantly flout this hoary hortation of the punters' poet, Sumner A Ingmark:

If you disdain the technicalities,
Prepare for the eventualities.

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Best of Alan Krigman
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.