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Best of Alan Krigman

Gaming Guru

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When Dealers Make 21, It's Unbelievable; When You Do, It's Skill

7 August 1995

en's the last time this happened to you?

You're playing blackjack at a $10 table. You've had several winning hands in a row. You're $100 ahead. The high cards are coming thick and fast. You've "pressed" your bet up to $25.

The dealer sends you a pair of nines and draws a six-up. Your heart beats faster as you slide out another $25 chip and split your nines. You pull a two on the first nine, anxiously shove out another $25 to double-down, and draw an eight for a total of 19. You're strong. The dealer drops a third nine on the other half of your pair. You excitedly move out one more $25 chip and re-split (you're at an enlightened casino where this option is offered). You draw a 10 on the first for another stalwart 19 and an ace on the second for an even more potent 20.

Your whole $100 profit is on the line and you're sitting pretty with two 19s and a 20 against a vulnerable dealer six-up. You're congratulating yourself on your skillful play. Play that will pump up, probably even double, your already respectable lead. You're feeling good. The dealer flips over an eight in the hole for a 14. You're feeling even better. The dealer turns a two for a total of 16. You're counting your money. The moment of truth. The dealer pulls a five for a total of 21, sweeping away all your bets, wiping out your profit. "Unbelievable," you think to yourself. "Unbelievable," mutter two solid citizens who supposed they were secure with their 20s.

No, not unbelievable at all.

Yes, six is a dealer's weakest up-card. The chance of the dealer breaking with a six -- what everyone expected -- is 42.3 percent. Compare this with the chance of hitting 21 -- what everyone thought was unbelievable -- which turns out to be 9.7 percent. Now, 42.3 out of 100 is a reasonable probability, yet it's still less than a 50-50 proposition. And, it's only somewhat above four times more likely than the small -- but by no means insignificant -- 9.7 out of 100 probability of a 21.

For every up-card, the accompanying table gives the probability of the dealer getting an ace-10 blackjack and of drawing to a three-or-more-card 21. For reference, the chance of breaking for each up-card is also shown.

Ask any ten players how they think the chances of a dealer drawing a non-blackjack 21 change as the up-card gets higher -- for instance, five rather than three or nine rather than six. Unless you happen upon a mob of mathematicians leaving a calculator conclave, you're apt to get mostly wrong answers. You can see for yourself, from the table, that except for a reversal on 10 and ace, the dealer is more likely to get a non-blackjack 21 with smaller up-cards.

From the table, you can also see how the probability of the dealer drawing a 21 generally supports the strength of a player's position against weak up-cards. Not only does the chance the dealer will break rise from 35.4 to 42.3 percent as up-cards go from two through six, but the likelihood that a hand will be "made" at 21 falls from 11.8 to 9.7 percent.

Slight probability differences like these illustrate why blackjack became much more sophisticated after computers were applied to determine the optimum rules embodied in "basic strategy." Just assuming the "unseen" card is 10 is overly simplistic. Small shifts in percentages, for instance, indicate that it's advantageous to double-down on soft 17 (ace-six) against dealer three through six, but on a soft 15 (ace-four) against only four through six, and on soft 13 (ace-two) against just five and six. But it's the same small differences that leave room for luck, explaining why rookies sometimes break the bank while skilled players lick their wounds. Sumner A Ingmark, the blackjack bettors' Bobby Burns, may have put it best:

Novice beginners who go by their gut'll,
Sometimes do well without knowing rules subtle;
Evidence long-term, though, serves as rebuttal.

Up-card Probability
of blackjack
Probability of
non-blackjack 21
Probability
of breaking
2 0.0% 11.8% 35.4%
3 0.0 11.5 37.4
4 0.0 11.1 39.4
5 0.0 10.8 41.6
6 0.0 9.7 42.3
7 0.0 7.4 26.2
8 0.0 6.9 24.5
9 0.0 6.1 22.8
10 7.7 3.4 21.2
A 30.8 5.4 11.5
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.