The Care and Feeding of the Doubling Cube, Part 4
Danny Kleinman, 1980
Vision Laughs at Counting, Vol 2, © 1980 Danny Kleinman

# The Match Score Charts

The next few pages contain charts indicating equities and cube actions in a match to 9 points between opponents of equal skill. Not that there is no specific probability of winning which justifies doubling prior to the last shake (or virtual last shake). This phase of cube handling — doubling — depends on your opponent’s actual take point.

You can apply the charts for a 9-point match to matches of other lengths. For example, a 4–3 lead in a 7-point match is exactly equivalent to a 6–5 leading in a 9-point match.

As a benchmark for comparison, you may think of 0.25 as a normal minimum take and 0.50 as a normal minimum for doubling on the last shake. Probabilities higher than these in the charts demand conservative cube handling, and lower probabilities justify liberal cube actions.

You may find it worth your while to master these charts. But you may find it too much work. If so, what you give up by simply understanding the text and using the shortcuts to make estimates will be relatively little.

Probability of Winning a 9-Point Match
(Player’s score down side, opponent’s score across top)
* = Crawford game
 8 8* 7 6 5 4 3 2 1 0 8 .500 0.51 0.7 0.71 0.83 0.835 0.902 0.905 *8 .500 0.7 0.75 0.831 0.854 0.903 0.917 0.944 .952 7 .490 .300 0.5 0.597 0.677 0.746 0.806 0.845 0.88 .905 6 .300 .250 0.403 0.5 0.588 0.657 0.732 0.78 0.822 .854 5 .190 .169 0.323 0.412 0.5 0.58 0.655 0.711 0.764 .804 4 .170 .146 0.254 0.343 0.42 0.5 0.577 0.638 0.696 .742 3 .165 .097 0.194 0.268 0.345 0.423 0.5 0.565 0.628 .680 2 .098 .083 0.155 0.22 0.289 0.362 0.435 0.5 0.563 .618 1 .095 .056 0.12 0.178 0.236 0.304 0.372 0.437 0.5 .558 0 .048 0.095 0.146 0.196 0.258 0.32 0.382 0.442 .500

Gammon Price List
(Fraction of a win that a gammon is worth to the player)
At 1-level
 8 8* 7 6 5 4 3 2 1 0 7 1 1.00 0.75 1 0.72 0.82 0.62 0.75 0.57 0.67 6 0.02 .00 0.8 0.79 0.86 0.68 0.65 0.64 0.64 0.57 5 0.63 1.00 0.41 0.55 0.51 0.57 0.49 0.52 0.52 0.57 4 0.03 .00 0.45 0.56 0.56 0.48 0.5 0.51 0.46 0.46 3 0.71 1.00 0.44 0.46 0.53 0.52 0.51 0.53 0.52 0.54 2 0.03 .00 0.54 0.66 0.6 0.57 0.56 0.56 0.53 0.53 1 0.68 1.00 0.39 0.48 0.5 0.48 0.5 0.51 0.52 0.53 0 0.03 .00 0.49 0.51 0.59 0.54 0.55 0.54 0.53 0.52

At 2-level
 8 7 6 5 4 3 2 1 0 6 1.00 0.43 0.5 0.39 0.41 0.31 0.32 0.26 0.28 5 1.04 1 0.94 0.91 0.76 0.63 0.58 0.53 0.49 4 .67 0.74 0.71 0.73 0.63 0.55 0.49 0.5 0.45 3 .69 0.55 0.59 0.58 0.53 0.49 0.47 0.44 0.42 2 .76 0.59 0.6 0.63 0.56 0.54 0.51 0.48 0.46 1 .70 0.66 0.68 0.69 0.64 0.59 0.56 0.53 0.51 0 0.64 0.72 0.68 0.64 0.59 0.57 0.55 0.53

At 4-level
 7 6 5 4 3 2 1 0 4 0.43 0.33 0.2 0.21 0.15 0.14 0.11 0.11 3 1 0.68 0.48 0.39 0.32 0.26 0.22 0.2 2 1.48 1 0.7 0.6 0.46 0.39 0.33 0.3 1 2.1 1.43 1 0.8 0.64 0.54 0.45 0.39 0 1.76 1.18 0.98 0.78 0.68 0.53 0.5 0.48

At 8-level
 6 5 4 3 2 1 0 0 0.33 0.2 0.17 0.11 0.09 0.06 0.05

Money Games, All Levels:  0.50
(gammons half as important as the game)

Winning Probability Needed to Turn Cube on Last Shake
To 2-level
 7 6 5 4 3 2 1 0 7 0.5 0.44 0.36 0.35 0.42 0.42 0.41 0.42 6 0.56 0.5 0.39 0.45 0.42 0.51 0.43 0.47 5 0.64 0.61 0.5 0.5 0.52 0.54 0.49 0.51 4 0.65 0.55 0.5 0.5 0.51 0.53 0.52 0.54 3 0.58 0.58 0.48 0.49 0.5 0.51 0.49 0.5 2 0.58 0.49 0.46 0.47 0.49 0.5 0.49 0.5 1 0.59 0.57 0.51 0.48 0.51 0.51 0.5 0.5 0 0.58 0.53 0.49 0.46 0.5 0.5 0.5 0.5

To 4-level
 6 5 4 3 2 1 0 6 0.5 0.7 0.71 0.66 0.65 0.72 0.72 5 0.3 0.5 0.49 0.48 0.52 0.56 0.58 4 0.29 0.51 0.5 0.49 0.53 0.56 0.58 3 0.34 0.52 0.51 0.5 0.54 0.57 0.58 2 0.35 0.48 0.47 0.46 0.5 0.54 0.55 1 0.28 0.44 0.44 0.43 0.46 0.5 0.52 0 0.28 0.42 0.42 0.42 0.45 0.48 0.5

To 8-level
 4 3 2 1 0 4 0.5 0.72 0.81 0.88 0.88 3 0.28 0.5 0.63 0.74 0.77 2 0.19 0.37 0.5 0.62 0.66 1 0.12 0.26 0.38 0.5 0.56 0 0.12 0.23 0.34 0.44 0.5

Money Games, All Levels:  0.50
(the favorite should double on the last shake)

Minimum Winning Probability Needed to Take Cube
You must adjust for gammon threats before using these charts.
Where a range is given, the higher probability is needed to take
when the cube is useless to you, as on the last shake of the game.
At 2-level
 7 6 5 4 3 2 1 0 8 .50 .02 0.39 0.03 0.41 0.03 0.41 7 .30 .29 0.19 0.2 0.21 0.24 0.2 .23 6 .25–.36 .31 0.23 0.25 0.22 0.29 0.23 .24 5 .17–.34 .32–.36 0.25 0.26 0.26 0.28 0.25 .27 4 .21–.36 .31 0.26 0.25 0.26 0.28 0.25 .27 3 .19–.30 .31 0.24 0.25 0.25 0.27 0.25 .26 2 .21–.33 .28 0.25 0.25 0.25 0.26 0.25 .26 1 .17–.29 .29–.30 0.26 0.24 0.26 0.26 0.25 .26 0 .19–.31 .00 0.26 0.23 0.26 0.26 0.26 .25

At 4-level
 6 5 4 3 2 1 0 7 .30 .50 .42 .35 0.37 0.4 0.39 6 .25 .40 .33 .31 0.31 0.34 0.36 5 .17 .32 .29 .26 0.29 0.31 0.31 4 .15–.19 .25–.31 .28 .26 0.27 0.3 0.31 3 .10–.16 .19–.29 .26 .25 0.27 0.29 0.3 2 .08–.17 .15–.26 .22–.24 .33 0.25 0.27 0.28 1 .06–.14 .12–.24 .18–.23 .22 0.24 0.26 0.27 0 .05–.14 .11–.23 .15–.22 .20–.21 0.23 0.25 0.26

At 8-level
 4 3 2 1 0 6 .25 0.4 .50 .60 0.54 5 .17 0.32 .41 .50 0.49 4 .15 0.25 .34 .42 0.41 3 .10 0.19 .27 .35 0.36 2 .08 0.16 .22 .29 0.3 1 .06 0.12 .18 .24 0.26 0 .05–.06 0.1 .15–.16 .20–.23 0.23