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last roll situation
Posted By: scotty
Date: Tuesday, 8 July 2014, at 3:23 a.m.
In Response To: last roll situation (scotty)
Wow, such an incredible learning experience! Initially, I failed to see that the trailers cube would allow him to collect enough additional wins to justify offering the cube. Following my first attempt at this problem, where I felt that the leader would cash out, I thought about all those wins left on the table, and thought about CPW, not realizing there was a way to collect those wins.
Do other situations arise where killing the cube is the optimal strategy?
To answer the questions posed, the trailer wins 40.2%, and the trailer must cube to collect those wins. Hoping that i got it right this time! Thanks so much!
Long answer follows!
the trailer clears on his first roll with 66, 55, 44, 33, and 65 (6/36)
the leader gets a chance to roll 30/36, and clears with 22 33 44 55 66 and 20 others, 24/36
the leader wins on his first roll 5/6 x 2/3 = 5/9
the following first rolls for the trailer do not guarantee that he will be off in 2 rolls:
61, 51, 41, 31, 21, 11 32, 42, (16 rolls)
the trailer has 20 rolls which guarantee he will be off on his next roll.
first rolls for the trailer, and number of misses on the second roll
61, one checker remaining on 4pt. 2
51, one checker remaining on 4pt. 2
41, one checker remaining on 5pt. 5
31 checkers on 1 and 5 pts. 11
21 checkers on 2 and 5 pts. 17
11 checkers on 1 and 5 pts. 11
32, one checker remaining on 5pt. 5
42, checkers on 1 and 3 pts. 2
total misses on second roll 55
the trailer misses 55/1296
the leader wins an additional 55/1296 games for total wins of 59.8%
the trailer has enough winning chances to cube if the game is played out, and if he does cube, the cube is dead and the game is played to fruition. If the trailer fails to cube, then we have already seen that the leader can cube, and the trailer will have to drop. The trailer's best action is to cube, and the leader should take.

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