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Best of Alan Krigman

Gaming Guru

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Pinning numbers on the hopes and fears of blackjack

12 December 2011

Many casino games are characterized by multiple moments of truth. Craps has one whenever the shooter throws the dice. At blackjack, they occur every time a card is turned over – yours or the dealer’s. When you take whatever action you do, you could bust, losing then and there. If not, you’ll finish with a total between 11 and 21 and nervously await the dealer’s fate. Anxiously but with a glimmer of optimism if you’re near the low end of this regime, or confidently yet with trepidation if you’re in the high range. The balance between these conflicting feelings depends not only on your own final total but also on the weakness or strength of the dealer’s upcard.

The math mavens can pin numbers on those hopes or fears for any given situations. The usual parameter employed for this purpose is what the statisticians call “expectation.” Its value can be envisioned as the average percentage of a wager which would be won or lost over a statistically large set of trials. For example, the expectation for a player’s final total of 18 against a dealer’s seven-up is +40.0 percent – an average profit of 40.0 cents per dollar at risk.

Expectation offers the benefit of being a single figure that describes a solid citizen’s prospects. By convention, it’s negative when the house has the edge and positive when the bettors are favored. The magnitude, generally on a scale from -100 to +100 percent, denotes the strength of the tendency either way. Expectation is a considerably more useful gauge to the bosses than the bettors since the former reckon their performance over millions of decisions while the latter rarely get to the stage where averages outweigh volatility and reliably anticipate actual results.

Players often find likelihoods of winning, losing, or pushing to be more intuitively satisfactory predictors of how they’ll fare than expectations. In some games, these chances can be determined simply by counting the possible outcomes. For instance, a bet on a three-number row at double-zero roulette has three ways out of 38 to win (7.9 percent), 35 out of 38 to lose (92.1 percent), and zero out of 38 to push. The prospects associated with the final hands at blackjack are more complicated to calculate. The probabilities, along with the expectations, are given in the accompanying tables for playable hands – those excluding dealer blackjacks with 10- or ace-up.

 
upcard   Player wins with final total:					
           <17      17      18      19      20      21
2         35.35%  35.35%  49.32%  62.77%  75.77%  88.17%
3         37.41%  37.41%  50.86%  63.91%  76.44%  88.51%
4         39.55%  39.55%  52.60%  65.05%  77.18%  88.83%
5         41.79%  41.79%  53.98%  66.22%  77.98%  89.20%
6         42.29%  42.29%  58.86%  69.48%  80.12%  90.28%
7         26.20%  26.20%  63.11%  76.90%  84.75%  92.61%
8         24.40%  24.40%  37.28%  73.26%  86.13%  93.05%
9         22.90%  22.90%  34.93%  46.73%  81.89%  93.91%
10        23.01%  23.01%  35.13%  47.22%  59.33%  96.24%
Ace       16.69%  16.69%  35.52%  54.43%  73.33%  92.25%


upcard	Player loses with final total:					
           <17      17      18      19      20      21
2         64.65%  50.68%  37.23%  24.23%  11.83%  0.00%
3         62.59%  49.14%  36.09%  23.56%  11.49%  0.00%
4         60.45%  47.40%  34.95%  22.82%  11.17%  0.00%
5         58.21%  46.02%  33.78%  22.02%  10.80%  0.00%
6         57.71%  41.14%  30.52%  19.88%  9.72%   0.00%
7         73.80%  36.89%  23.10%  15.25%  7.39%   0.00%
8         75.60%  62.72%  26.74%  13.87%  6.95%   0.00%
9         77.10%  65.07%  53.27%  18.11%  6.09%   0.00%
10        76.99%  64.87%  52.78%  40.67%  3.76%   0.00%
Ace       83.31%  64.48%  45.57%  26.67%  7.75%   0.00%


upcard	Player pushes with final total:					
           <17      17      18      19      20      21
2          0.00%  13.97%  13.45%  12.99%  12.40%  11.83%
3          0.00%  13.45%  13.05%  12.53%  12.07%  11.49%
4          0.00%  13.05%  12.45%  12.13%  11.65%  11.17%
5          0.00%  12.19%  12.24%  11.76%  11.21%  10.80%
6          0.00%  16.56%  10.62%  10.64%  10.16%  9.72%
7          0.00%  36.90%  13.79%  7.85%   7.87%   7.39%
8          0.00%  12.88%  35.98%  12.87%  6.93%   6.95%
9          0.00%  12.02%  11.80%  35.16%  12.03%  6.09%
10         0.00%  12.11%  12.09%  12.12%  36.90%  3.76%
Ace        0.00%  18.83%  18.92%  18.89%  18.93%  7.75%


upcard	Player's expectation with final total:					
           <17      17      18      19      20      21
2        -29.29%  -15.32%  12.10%  38.54%  63.94%  88.17%
3        -25.18%  -11.73%  14.77%  40.36%  64.96%  88.51%
4        -20.91%   -7.85%  17.66%  42.24%  66.01%  88.83%
5        -16.42%   -4.23%  20.21%  44.21%  67.18%  89.20%
6        -15.42%    1.15%  28.33%  49.59%  70.39%  90.28%
7        -47.59%  -10.69%  40.01%  61.64%  77.36%  92.61%
8        -51.21%  -38.32%  10.54%  59.39%  79.18%  93.05%
9        -54.19%  -42.17% -18.35%  28.61%  75.80%  93.91%
10       -53.98%  -41.86% -17.66%  6.55%   55.57%  96.24%
Ace      -66.62%  -47.79% -10.05%  27.76%  65.58%  92.25%

To illustrate how to interpret the data, say you finish with 19 against an eight-up. Your chances are 73.26 percent of winning, 13.87 percent of losing, and 12.87 percent of pushing. Expectation is 59.39 percent – an average profit of 59.39 cents per dollar bet. The tables also show that the worst total, anything between 11and 16, always has a shot. And, conversely, the best total, 21, can’t lose but may push and therefore isn’t necessarily a winner. Such situations bring to mind this veritable verse of that praiseworthy punters’ poet, Sumner A Ingmark:

When there’s a chance, however small,
Don’t rush to celebrate your haul,
Or prematurely grieve your fall.
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.