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BGonline.org Forums
bearoff formula for around 20 pips
Posted By: Bob Koca In Response To: bearoff formula for around 20 pips (John O'Hagan)
Date: Wednesday, 14 November 2012, at 6:52 p.m.
You may need to estimate the chances of being off in 1,2,3,4 moves and calculate from there.
There are some very surprising counterintuitvie things happening in short bearoffs that will foil commonly used estimation techniques. In particular do not rely on the Kleinmann technique. Suppose we change a symmetric 70 -70 pip race into a 69 - 69 pip race. The race is a bit shorter giving the underdog player not on turn a little less win chance. But consider symmetric 4 checker vs. 4 checker bearoffs. The average win % for the on turn player is:
11 pips: .7258
12 pips: .7342
13 pips: .7349
14 pips: .7404
15 pips: .7424
For that range making the race longer hurts the trailer. What bearoff estimation technique would capture that aspect?
What is going on? If you wnat to think about it on your won first combacvk to this point later.
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What is going on is that most of the chances of being off in a certain number of moves goes to being off in exactly 2 or exactly 3. The chance of being off in one is small because big doubles are needed and the chance of being off in four is small because the pipcount is small enough. Supposing that all of the chances were being off in 2 (call it p) or 3 (then 1 - p) the not on turn player's win chance is (1-p)p. This is maximized for p = .5 The shorter races are more toward that optimal situation for the not on turn player.
As an example the 11 pip position 120001 is off
in 1 turn with 2.78%
in 2 turns 48.23%
in 3 turns 48.8%
in 4 turns 0.1%
and vs itself the on turn player wins 73.62%
Making the race longer mostly transfers some of the chance of being off in exactly 2 to the chance for being off in 3 and that helps the leader. There is some increase in the chance of being off in 4 also which helps the trailer (giving another way he can win) but that is not as important an increase in that pips range.
Compare to 200002 which is particularly good for the leader. It is the best win % for the leader of symmetric 4 checker bearoffs from 11 to 15 pips. It bears off:
in 1 turn 2.78%
in 2 turns 20.91%
in 3 turns 71.14%
in 4 turns 5.17%
and vs. itself the on turn player wins 77.7%
This is particularly good because it is an inefficient 14 pips making a a especially large swing of chances from off in 2 to off in 3. The inefficiency isn't as important though to the chances of taking 4 rolls because the 1's are still efficient.
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