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# How good is your total against various dealer upcards at blackjack?

29 October 2012

Say you’ve played out your own hand at blackjack. The dealer didn’t have a blackjack (or the round would have ended before it began), and you didn’t bust. You’re therefore at a moment of truth, anticipating how the dealer will fare.

If your total is 21, you’re in fine shape. You can’t lose. At worst, you can push. The probability of pushing is least (3.76 percent), so your prospects of winning are best (96.24 percent), when the dealer has 10-up. Expectation for even-money bets equals the chance of winning minus that of losing; since 21 can’t lose, expectation and chance of winning are numerically the same – a positive 96.24 percent, average earnings of 96.24 cents per dollar at risk over many instances of this situation. The chance of a push is highest (11.83 percent), so your prospects of winning and expectation for profit are lowest (88.17 percent), when the dealer has 2-up. Intermediate positions, from strongest to weakest, occur for 9-, 8-, 7, ace-, 6-, 5-, 4-, and 3-up.

You’re also sitting pretty with totals of 20. But maybe not as pretty as you think. A 20's best chance of winning is against 8-up – 86.13 percent. Expectation is greatest against this upcard as well, but owing to possible pushes it’s less – 79.18 percent. Chance of winning and expectation are both least against a dealer’s 10-up, at 59.33 percent and 55.57 cents per dollar up for grabs, respectively, but this bad news results more from pushes than losses. The accompanying table gives the probabilities of winning, pushing, and losing as well as the expectation for a player’s 20 against all upcards. The data show why solid citizens with 20s shouldn’t count their money before the dealer acts, especially with 2- to 6-up which almost everyone considers busts-in-waiting. In fact, dealers are more apt to reach 21 and edge out player’s 20s with 2- through 6- than with 7- through ace-up.

Probabilities of winning, pushing, and losing with player’s 20
against various upcards, and resulting expectations

```up-card     probabilities        expectation
win    push    lose
2       5.77%  12.40%  11.83%      63.94%
3      76.44%  12.07%  11.49%      64.96%
4      77.18%  11.65%  11.17%      66.01%
5      77.98%  11.21%  10.80%      67.18%
6      80.12%  10.16%   9.72%      70.39%
7      84.75%   7.87%   7.39%      77.36%
8      86.13%   6.93%   6.95%      79.18%
9      81.89%  12.03%   6.09%      75.80%
10     59.33%  36.90%   3.76%      55.57%
Ace	73.33% 18.93%   7.75%      65.58%
```

Bettor 19s have less chance of winning than 20s or 21s – from highs of 76.90 and 73.26 percent against 7- and 8-up, to lows of 47.22 and 46.73 percent against 9- and 10-up, respectively. Expectation is always positive, though, running from a maximum of 61.64 percent against 7-up to a minimum of 6.55 percent against 10-up.

Blackjack buffs typically overestimate the strength of 18s. Chances of this total losing exceed 50 percent and expectations are accordingly negative against 9- or 10-up; the probability of 18 losing against ace-up is less, 45.57 percent, but expectation is negative here, too. An 18 is most promising against 7-up, with a probability of winning equal to 63.11 percent and an expectation of earning an average of 40.01 percent.

A 17 is, arguably, the most frustrating stopping point. Basic Strategy is to stand on hard 17, so players who finish at that level often blithely believe they’ve squeezed by. The mistake is that, like totals of 16 or less, a 17 can’t win unless the dealer busts. It differs from lower values only because it has a chance of pushing and not losing if the dealer also gets 17. Probabilities of winning are below 50 percent for all upcards and expectation is negative except against 6-up, where it’s a paltry 1.15 percent average profit on the dollar. The worst upcard to fight with 17 is an ace, where expectation is to lose an average of 47.79 percent – followed by 9-, 10-, and 8-up.

With totals from 12 through 16 against dealer’s 2- through 6-up, Basic Strategy is to stand. Not that it’s profitable owing to the dealer being apt to bust; rather, because it’s riskier to hit than stand and lose because the dealer made 17 through 21. In these situations, players’ highest hopes are against 6-up which has a 42.29 percent chance of busting, decreasing from there through 2-up where the dealer’s probability of breaking drops to 35.35 percent.

Given that the probability of winning and the expectation aren’t equal other than for player 21s, enquiring minds should want to know whether the maxima for the two the occur with different upcards for any totals. The answer is “no,” they always coincide. A phenomenon foretold by this vague yet veracious verse by the venerable virtuoso of casino cadence, Sumner A Ingmark:

‘Though winning and losing don’t cover all bases,
Don’t count on “no action” to overturn races.