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Rollouts of Opening Moves
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Rollout Settings
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The rollouts on this page where performed using Gnu Backgammon, version 0.14, which is the latest and strongest version of Gnu BG as of early 2006. The rollouts were performed using 2-ply play, which means that Gnu BG looked ahead two rolls as it made each play during the rollouts, averaging the resulting 1296 future positions for each candidate to determine the best play as it played through each trial. This is the highest setting at which Gnu BG can perform rollouts in any reasonable time. In total, the rollouts on this page took over seven months to complete.
Each play was rolled out 46,656 times, which is 36 × 36 × 36, thus guaranteeing that each possible combination of the first three rolls of each play is equally represented. Variance reduction was used to reduce the effect of random error from lucky sequences of dice. The standard error when performing this number of trials using variance reduction is less than 0.001. Standard errors this small are usually (but not always!) enough to separate the best play from the others. A few candidates were rolled out less often than this (between 5,000 and 20,000 times each) when it was clear they would be nowhere near the best in any type of match situation.
Rollouts were performed using cubeless money play. Cubeless means that games are played without a doubling cube and always played to the end of the game. Gammons and backgammons count as usual.
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Reading the Charts
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The EQUITY column of each chart shows the estimated cubeless equity of each candidate play. The best play is displayed with a white background and the other plays are displayed with progressively darker backgrounds to give a rough idea of the amount of equity lost. Plays that give up more than 0.05 in cubeless equity are not listed.
The check marks  denote the best play in each of the following situations:
The DMP, GG, and GS values are inferred from the probabilities of the cubeless rollout. It is possible that a true rollout (according to score) would give different results than those shown for these special scores.
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Rollouts
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PLAY |
W G BG |
EQUITY |
CL |
DMP |
GG |
GS |
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13/11, 6/5
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+ .5058, .1415, .0059
− .4942, .1418, .0084
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+.0088
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(.0074)
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24/23, 13/11
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+ .4996, .1369, .0056
− .5004, .1319, .0053
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+.0045
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(.0043)
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(.0124)
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(.0216)
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24/21
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+ .4918, .1254, .0056
− .5082, .1315, .0057
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−.0226
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(.0314)
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(.0280)
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(.0602)
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(.0148)
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13/10
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+ .4897, .1378, .0059
− .5103, .1419, .0068
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−.0256
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(.0344)
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(.0322)
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(.0396)
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(.0398)
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24/23, 24/22
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+ .4868, .1213, .0051
− .5132, .1329, .0053
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−.0382
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(.0470)
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(.0380)
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(.0784)
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(.0276)
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PLAY |
W G BG |
EQUITY |
CL |
DMP |
GG |
GS |
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8/5, 6/5
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+ .5546, .1708, .0071
− .4454, .1150, .0051
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+.1670
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PLAY |
W G BG |
EQUITY |
CL |
DMP |
GG |
GS |
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13/11, 13/10
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+ .4977, .1493, .0065
− .5023, .1387, .0072
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+.0053
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(.0080)
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(.0172)
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24/21, 13/11
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+ .5017, .1355, .0064
− .4983, .1341, .0060
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+.0052
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(.0001)
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(.0196)
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24/22, 13/10
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+ .4985, .1366, .0059
− .5015, .1345, .0056
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−.0006
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(.0059)
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(.0064)
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(.0238)
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(.0072)
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13/10, 6/4
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+ .4918, .1369, .0061
− .5082, .1455, .0090
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−.0279
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(.0332)
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(.0198)
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(.0366)
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(.0426)
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24/22, 24/21
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+ .4924, .1223, .0054
− .5076, .1350, .0055
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−.0280
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(.0333)
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(.0186)
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(.0646)
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(.0204)
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13/8
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+ .4887, .1344, .0067
− .5113, .1469, .0064
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−.0348
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(.0401)
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(.0260)
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(.0478)
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(.0516)
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PLAY |
W G BG |
EQUITY |
CL |
DMP |
GG |
GS |
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24/23, 13/9
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+ .4984, .1405, .0058
− .5016, .1348, .0059
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+.0024
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(.0032)
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(.0050)
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13/9, 6/5
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+ .5000, .1414, .0062
− .5000, .1448, .0091
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−.0063
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(.0087)
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(.0168)
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24/23, 24/20
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+ .4886, .1223, .0052
− .5114, .1320, .0045
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−.0318
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(.0342)
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(.0228)
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(.0610)
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(.0140)
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13/8
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+ .4887, .1344, .0067
− .5113, .1469, .0064
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−.0348
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(.0372)
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(.0226)
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(.0366)
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(.0436)
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24/20, 6/5
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+ .4879, .1270, .0059
− .5121, .1429, .0064
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−.0406
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(.0430)
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(.0242)
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(.0530)
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(.0372)
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PLAY |
W G BG |
EQUITY |
CL |
DMP |
GG |
GS |
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8/4, 6/4
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+ .5380, .1697, .0074
− .4620, .1242, .0055
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+.1234
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PLAY |
W G BG |
EQUITY |
CL |
DMP |
GG |
GS |
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13/10, 13/9
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+ .4984, .1529, .0075
− .5016, .1374, .0073
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+.0125
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(.0024)
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(.0118)
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24/21, 13/9
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+ .4996, .1404, .0066
− .5004, .1342, .0060
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+.0060
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(.0065)
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(.0226)
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(.0030)
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24/20, 13/10
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+ .4989, .1355, .0061
− .5011, .1330, .0056
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+.0008
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(.0117)
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(.0014)
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(.0338)
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(.0020)
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24/21, 24/20
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+ .4980, .1223, .0056
− .5020, .1311, .0048
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−.0120
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(.0245)
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(.0032)
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(.0620)
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PLAY |
W G BG |
EQUITY |
CL |
DMP |
GG |
GS |
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24/23, 13/8
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+ .5006, .1350, .0054
− .4994, .1311, .0049
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+.0056
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13/8, 6/5
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+ .4976, .1374, .0058
− .5024, .1348, .0067
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−.0031
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(.0087)
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(.0060)
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(.0012)
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(.0134)
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24/18
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+ .4930, .1179, .0048
− .5070, .1355, .0053
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−.0321
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(.0377)
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(.0152)
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(.0494)
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(.0240)
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PLAY |
W G BG |
EQUITY |
CL |
DMP |
GG |
GS |
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24/22, 13/8
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+ .5007, .1334, .0058
− .4993, .1327, .0047
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+.0032
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(.0142)
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13/11, 13/8
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+ .4970, .1442, .0074
− .5030, .1414, .0065
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−.0023
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(.0055)
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(.0074)
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(.0248)
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13/8, 6/4
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+ .4899, .1335, .0057
− .5101, .1403, .0063
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−.0276
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(.0308)
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(.0216)
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(.0356)
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(.0368)
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PLAY |
W G BG |
EQUITY |
CL |
DMP |
GG |
GS |
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8/3, 6/3
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+ .5173, .1617, .0076
− .4827, .1339, .0062
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+.0638
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24/21, 13/8
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+ .5061, .1386, .0071
− .4939, .1338, .0045
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+.0196
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(.0442)
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(.0224)
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(.0686)
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(.0222)
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13/10, 13/8
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+ .4970, .1498, .0073
− .5030, .1406, .0062
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+.0042
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(.0596)
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(.0407)
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(.0645)
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(.0540)
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PLAY |
W G BG |
EQUITY |
CL |
DMP |
GG |
GS |
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24/20, 13/8
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+ .5051, .1362, .0072
− .4949, .1331, .0050
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+.0155
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(.0170)
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(.0002)
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13/9, 13/8
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+ .4999, .1499, .0077
− .5001, .1365, .0064
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+.0145
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(.0010)
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(.0104)
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(.0174)
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24/15
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+ .5034, .1168, .0052
− .4966, .1313, .0046
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−.0071
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(.0226)
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(.0034)
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(.0592)
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13/4
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+ .4852, .1382, .0066
− .5148, .1426, .0070
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−.0344
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(.0499)
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(.0398)
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(.0528)
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(.0590)
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PLAY |
W G BG |
EQUITY |
CL |
DMP |
GG |
GS |
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13/7, 8/7
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+ .5353, .1516, .0063
− .4647, .1201, .0049
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+.1035
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PLAY |
W G BG |
EQUITY |
CL |
DMP |
GG |
GS |
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24/18, 13/11
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+ .5068, .1270, .0058
− .4932, .1346, .0052
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+.0066
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(.0028)
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13/5
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+ .4931, .1421, .0067
− .5069, .1390, .0069
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−.0109
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(.0175)
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(.0274)
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(.0362)
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24/16
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+ .5028, .1187, .0053
− .4972, .1363, .0050
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−.0117
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(.0183)
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(.0080)
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(.0274)
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(.0114)
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PLAY |
W G BG |
EQUITY |
CL |
DMP |
GG |
GS |
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24/18, 13/10
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+ .5053, .1299, .0062
− .4947, .1339, .0056
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+.0072
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(.0014)
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24/15
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+ .5034, .1168, .0052
− .4966, .1313, .0046
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−.0071
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(.0143)
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(.0038)
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(.0300)
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24/21, 13/7
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+ .4863, .1346, .0062
− .5137, .1403, .0063
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−.0332
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(.0404)
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(.0380)
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(.0286)
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(.0522)
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13/4
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+ .4852, .1382, .0066
− .5148, .1426, .0070
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−.0344
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(.0416)
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(.0402)
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(.0236)
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(.0590)
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13/10, 13/7
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+ .4816, .1405, .0077
− .5184, .1489, .0087
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−.0462
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(.0534)
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(.0474)
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(.0262)
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(.0788)
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PLAY |
W G BG |
EQUITY |
CL |
DMP |
GG |
GS |
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8/2, 6/2
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+ .4981, .1540, .0077
− .5019, .1417, .0060
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+.0102
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(.0168)
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(.0376)
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24/18, 13/9
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+ .5034, .1348, .0061
− .4966, .1350, .0058
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+.0069
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(.0033)
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(.0062)
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(.0278)
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(.0136)
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24/14
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+ .5065, .1179, .0053
− .4935, .1313, .0046
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+.0003
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(.0099)
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(.0554)
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PLAY |
W G BG |
EQUITY |
CL |
DMP |
GG |
GS |
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24/13
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+ .5282, .1164, .0053
− .4718, .1200, .0036
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+.0545
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24/18, 13/8
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+ .5099, .1287, .0068
− .4901, .1350, .0045
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+.0158
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(.0387)
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(.0366)
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(.0120)
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(.0666)
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Summary
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Here are all the opening rolls ranked from best to worst. The best roll is with an equity of +.1670. The worst roll is with an equity of +.0024.
Your average equity if you win the opening roll is +.0393.
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PLAY |
EQUITY |
W G BG |
L G BG |
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8/5, 6/5
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+.1670
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+ .5546, .1708, .0071
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− .4454, .1150, .0051
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8/4, 6/4
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+.1234
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+ .5380, .1697, .0074
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− .4620, .1242, .0055
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13/7, 8/7
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+.1035
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+ .5353, .1516, .0063
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− .4647, .1201, .0049
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8/3, 6/3
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+.0638
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+ .5173, .1617, .0076
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− .4827, .1339, .0062
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24/13
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+.0545
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+ .5282, .1164, .0053
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− .4718, .1200, .0036
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24/20, 13/8
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+.0155
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+ .5051, .1362, .0072
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− .4949, .1331, .0050
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13/10, 13/9
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+.0125
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+ .4984, .1529, .0075
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− .5016, .1374, .0073
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8/2, 6/2
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+.0102
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+ .4981, .1540, .0077
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− .5019, .1417, .0060
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13/11, 6/5
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+.0088
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+ .5058, .1415, .0059
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− .4942, .1418, .0084
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24/18, 13/10
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+.0072
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+ .5053, .1299, .0062
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− .4947, .1339, .0056
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24/18, 13/11
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+.0066
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+ .5068, .1270, .0058
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− .4932, .1346, .0052
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24/23, 13/8
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+.0056
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+ .5006, .1350, .0054
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− .4994, .1311, .0049
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13/11, 13/10
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+.0053
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+ .4977, .1493, .0065
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− .5023, .1387, .0072
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24/22, 13/8
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+.0032
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+ .5007, .1334, .0058
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− .4993, .1327, .0047
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24/23, 13/9
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+.0024
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+ .4984, .1405, .0058
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− .5016, .1348, .0059
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| Average |
+.0393
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+ .5127, .1447, .0065
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− .4873, .1315, .0057
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Gammon Rate and Backgammon Rate
The totals in this table allow us to compute overall gammon and backgammon rates. We see that the gammon rate when Gnu Backgammon plays itself is .1447 + .1315 = .2762.
That is, 27.62% of all backgammon games end in a gammon if played to completion.
And the backgammon rate is .0065 + .0057 = .0122. That is, 1.22% of all backgammon games end in a backgammon if played to completion.
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Return to
: Backgammon Galore
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