Probability and Statistics

Forum Archive : 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)  [GammOnLine forum]
Calculating winning chances  (Douglas Zare, June 2000) 
Chance of rolling x doubles in y rolls  (Raccoon+, July 2007)  [GammOnLine forum]
Chance of rolling x or more pips in y rolls  (Tom Keith, Feb 2004)  [GammOnLine forum] [Long message]
Clumping of random numbers  (Gary Wong, Sept 1998) 
Counting shots  (Koyunbaba+, June 2007) 
Counting shots  (John Little+, Mar 2007)  [GammOnLine forum]
Distribution of points per game  (Roland Sutter, June 1999) 
Distribution of points per game  (Stig Eide+, Sept 1995)  [Recommended reading]
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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