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How to estimate chances of hitting shot?

Posted By: Daniel Murphy
Date: Tuesday, 27 November 2012, at 10:01 p.m.

In Response To: How to estimate chances of hitting shot? (Mr Majestyk)

38 * 58.3%)/8.1667 = 2.7 rolls. Therefore blue's shot Equity is 8 * 2.7 = 21.6%

I think that calculation underestimates how long White might maintain the 11 point before volunteering or being forced to move from that point. The average roll of White's 21 rolls that do not blot or (all or nearly) clear is only 6.48 pips. So White can handle about 3 and a half of such rolls, on average, not 2.7, before being forced to play from the 11 point with one of them.

This suggests to me that if Blue wants to try to count immediate hits (8%) and future hits (fractions of 8%), he can reasonably count on getting up to four turns to get a shot. Besides White's free pip count of 38, the other limiting factor is Blue's free pip count, which is only 32. About 15% of the time, Blue will roll the 33rd pip that forces him to run one or both checkers from the 18 point before his fourth turn.

But I also think that the calculation's result overestimates the probability that Blue will hit a shot. We know that Blue's chance of hitting a shot on his first turn is 8.0246913580246913580246913580247%. If Blue gets up to four chances to hit, and 8/36 numbers leave a shot but don't blot thereafter (they rarely will), and 7/36 numbers clear (including 22 and 11 which nearly do) and 21/36 numbers maintain the 11 point vs. bar point formation, then Blue will hit

8% + 58.33% * 8% + 34.02% * 8% + 19.84% * 8% = 8.02% + 4.68% + 2.73% + 1.59% = 17.02% of the time. That's not the whole story, though. We should reduce that number by something, perhaps by about 1%, since sometimes Blue will already have run on his second or third roll. But we should also increase that number by (a) about 1%, perhaps, because sometimes Blue gets five or more chances to get a shot, plus (b) about 3% because 64 61 62 22 11 and 63 aren't "game over" -- they leave White 7, 4, 2, 2, 2, and 1 numbers that don't clear, for an average of 3.6/36 rolls that don't clear, for about a (3.6 * 11)/1296 chance of hitting.

I guess I get about 20% hit vig.

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