short bearoff/calculating

Setting Up for 22

The two best plays are 6/off 2/1 and 6/off 3/2. Blue does slightly better with the latter. Here is why:

After moving 3/2, no matter what White rolls, Blue will win outright when he rolls 22 on his next turn.
After moving 2/1, Blue can still win when he rolls 22, but only when White rolls a pair of non-doublets.
All other sequences transpose. In particular, when Blue rolls 11 or 21 on his next turn, it does not matter whether he plays 3/2 or 2/1 now.

After Blue Plays 6/off, 2/1

P( win on 1st turn )

The doublets 33, 44, 55, and 66 give Blue the win. As noted above, 22 does not. Blue gets these wins regardless of what White has rolled.

P( win on 1st turn ) = 4/36

P( win without trouble on 2nd turn )

By the time Blue gets his second turn—if he gets a second turn—White will have completed two turns himself. Only when White has failed to roll doublets will Blue get a second turn. By the same token, Blue won’t get a second turn if he wins by rolling doublets on his first turn.

Blue has a couple of trouble numbers on his 1st turn. They are 11 and 21. So long as he does not roll one of them on his first turn, he will win whenever he gets a second turn.

P( win on 1st turn ) = 4/36
P( trouble on 1st turn ) = 3/36
P( no trouble & no win ) = 29/36

P( win without trouble on 2nd turn )
= (5/6)*(29/36)*(5/6)
= 725/1296

Winning after Trouble

When Blue rolls 11 on his first turn, he will be left with a single checker on the 4pt on his second turn. If he can avoid a roll of 21, he will win.

When Blue rolls 21 on his first turn, he will leave checkers on his 1pt and 4pt. Subsequent rolls of 11, 21, 31, and 32 will cause him to lose. Otherwise, he wins when he rolls any of the other 29 numbers.

As above, we require that White does not roll a doublet on either of his turns.

P( win after trouble 11 )
= (5/6)*(1/36)*(5/6)*(34/36)
= 850/46656

P( win after trouble 21 )
= (5/6)*(2/36)*(5/6)*(29/36)
= 1450/46656

P( win after trouble ) = 2300/46656

Total Chance of Winning

The sum of the probabilities above gives the overall chance of winning.

P( win ) = P( win on 1st turn )
+ P( win without trouble on 2nd turn )
+ P( win after trouble )

= 4/36 + 725/1296 + 2300/46656
= 33584/46656
= 71.98%

After Blue Plays 6/off, 3/2

Calculations here are very similar. The only difference is that a roll of 22 moves into the winning category on the first turn. At the same time, it is removed from the winners on the second turn.

P( win on 1st turn ) = 5/36
P( trouble on 1st turn ) = 3/36
P( no trouble & no win ) = 28/36

P( win without trouble on 2nd turn )
= (5/6)*(28/36)*(5/6)
= 700/1296

P( win after trouble ) = 2300/46656

P( win ) = 5/36 + 700/1296 + 2300/46656
= 33980/46656
= 72.83%

What’s the Difference?

In his original post, Bob asks what the difference is between plays. The calculation makes it clear. You gain 1/36 by moving 22 into the group of winners on the first turn. You simulataneously lose 25/1296 by removing 22 from the winners on the second turn.

P( difference ) = 1/36 - 25/1296

Mike

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