Probability and Statistics

 Average luck of each roll

 From: Timothy Chow Address: tchow8@hotmail.com Date: 31 March 2013 Subject: What is the unluckiest roll, on average? Forum: BGonline.org Forums

```I have a vague feeling that this may have been discussed on BGO before but
I can't find it.

Suppose we got a bot to play a kajillion games of backgammon against
itself, and for each roll, we tallied its estimate of how lucky/unlucky
that roll was. What roll would be the unluckiest one, on average?

Clearly, doublets tend to be lucky. My gut feeling is that 52 or 51 tends
to be unlucky. It would be amusing to get some hard evidence on the
subject. Of course a bot rollout isn't necessarily the final word if we're
interested in what roll is the unluckiest in human-versus-human play, but
it should give a good indication.
```

 Phillippe Michel  writes: ```A relatively small sample (about 32000 rolls) gives : 51 -0.04133 21 -0.04033 41 -0.03075 32 -0.02303 61 -0.02021 63 -0.01971 43 -0.01754 42 -0.01538 52 -0.01402 64 -0.01235 62 -0.01104 31 -0.01058 65 -0.01043 53 -0.00285 54 -0.00028 22 +0.06238 11 +0.06289 33 +0.08236 66 +0.12256 44 +0.13023 55 +0.13927 ```

 Havard Raddum  writes: ```Interesting to see that all non-doublets have negative luck, on the average, and that the 6 doublets make up for that by all being quite lucky. For those who would like to reduce the luck element in backgammon, taking away the rule that doubles play twice would be a good place to start. ```

 Casper van der Tak  writes: ```But then every roll will be unlucky! ```

 Xavier  writes: ```Very similar results using XG (100,000 dice from XG2 vs XG1 unlimited session with Jacoby and Beaver) note that we did not reach Stats Sig for the worse roll the Gnu columns are from Philippe Michel post above for easy comparison I think it would be interesting to redo the study without racing positions. Extreme Gammon Gnu BG ------------------------------------- -------------- Dice Count Avg Luck Stddev Conf Dice Avg Luck 5-1 5632 -0.041 0.171 0.004 5-1 -0.041 2-1 5548 -0.039 0.178 0.005 2-1 -0.040 4-1 5567 -0.035 0.171 0.004 4-1 -0.031 3-2 5606 -0.032 0.174 0.005 3-2 -0.023 6-2 5728 -0.021 0.197 0.005 6-1 -0.020 4-3 5636 -0.016 0.178 0.005 6-3 -0.020 5-2 5372 -0.016 0.191 0.005 4-3 -0.018 6-3 5563 -0.013 0.199 0.005 4-2 -0.015 4-2 5698 -0.013 0.172 0.004 5-2 -0.014 6-1 5531 -0.012 0.198 0.005 6-4 -0.012 6-5 5680 -0.009 0.210 0.005 6-2 -0.011 5-4 5440 -0.009 0.181 0.005 3-1 -0.011 3-1 5657 -0.008 0.188 0.005 6-5 -0.010 5-3 5588 -0.005 0.188 0.005 5-3 -0.003 6-4 5570 -0.005 0.203 0.005 5-4 +0.000 1-1 2784 +0.058 0.268 0.010 2-2 +0.062 2-2 2672 +0.086 0.261 0.010 1-1 +0.063 3-3 2743 +0.109 0.282 0.011 3-3 +0.082 6-6 2676 +0.128 0.405 0.015 6-6 +0.123 5-5 2783 +0.131 0.354 0.013 4-4 +0.130 4-4 2659 +0.132 0.322 0.012 5-5 +0.139 ```

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### Probability and Statistics

Average game and match length  (JP White, Dec 2000)
Average luck of each roll  (Timothy Chow+, Mar 2013)
Average luck of each roll  (Jørn Thyssen+, Feb 2004)
Calculating winning chances  (Douglas Zare, June 2000)
Chance of rolling x doubles in y rolls  (Raccoon+, July 2007)
Chance of rolling x or more pips in y rolls  (Tom Keith, Feb 2004)
Clumping of random numbers  (Gary Wong, Sept 1998)
Counting shots  (Koyunbaba+, June 2007)
Counting shots  (John Little+, Mar 2007)
Distribution of points per game  (Roland Sutter, June 1999)
Distribution of points per game  (Stig Eide+, Sept 1995)
Expected variation in points after a series of games  (Achim Müller+, Feb 1999)
How many games to decide who's better?  (Stephen Turner, Mar 1997)
How often is too often?  (Gary Wong, Oct 1998)
Losing after bearing off 14 checkers  (Daniel Murphy, July 1999)
Number of games per match  (Jason Lee+, Jan 2005)
Number of rolls to enter x checkers from bar  (Michael Depreli+, Mar 2011)
Visualizing odds  (Daithi, Mar 2011)

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