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Last time we talked about some backgammon probabilities and today we'll see how those shortcuts can help us calculate correct cube decisions in bear-off positions.

This is our position from the last time:

is Player 2 score: 0 pip: 5 | ||||||||||||||||

Unlimited Game | ||||||||||||||||

pip: 3 score: 0 is Player 1 | ||||||||||||||||

XGID=-AA-------------------aa--:1:-1:-1:00:0:0:0:0:10 | ||||||||||||||||

on roll, cube action? |

It is the last roll position, because after throwing the dice either you or your opponent will win.

Cube decisions in the last roll positions are relatively simple:

- you should double if you are favorite to win the game (more than 50% chance, or 18 rolls)

- you should take if you can win at least 25% of the time (9 rolls)

To see why you need (only) 25% to take the cube, consider this:

If you play the same position 100 times and you drop the cube (offered on 2 value) every time you will lose 100x(-1) = -100 points.

If you play the position 100 times and you take the cube and manage to win 25% of the time, you will lose 75 times and win 25 times: 75x(-2) + 25x(+2) = -150 + 50 = -100 points.

If you can win more than 25% of the time, you will lose less by taking the cube than by passing, so you should take to minimize your losses.

Back to our position:

is Player 2 score: 0 pip: 5 | ||||||||||||||||

Unlimited Game | ||||||||||||||||

pip: 3 score: 0 is Player 1 | ||||||||||||||||

XGID=-AA-------------------aa--:1:-1:-1:00:0:0:0:0:10 | ||||||||||||||||

on roll, cube action? |

You will win this game when your opponent fails to bear off both his checkers.

That happens when he rolls a 1 on one die. Last time we said "When you need one number, 11 rolls are good". That is applicable here, too, and it is a much faster method than trying to see for every roll does it bear off or not.

So, 11 rolls fail to bear off both checkers (all ones). That means he will win on 36-11 = 25 rolls, and you will win on 11 rolls.

He is favorite to win the game so he must double, but you can win more than 25% of the time (9 rolls) so you have a take.

Consider this position:

is Player 2 score: 0 pip: 7 | ||||||||||||||||

Unlimited Game | ||||||||||||||||

pip: 3 score: 0 is Player 1 | ||||||||||||||||

XGID=-AA-----------------a--a--:1:-1:-1:00:0:0:0:0:10 | ||||||||||||||||

on roll, cube action? |

Here we can use second shortcut from the last time to help us calculate faster: "When you need two numbers, 20 rolls are good".

We can start by thinking he needs either 5 or 6 on his higher die. That's 20 winning rolls, per our shortcut.

But 61 and 51 don't bear off, that's 4 rolls (61, 16, 51, 15) which don't bear off. 20-4 = 16 winning rolls.

Now let's see how the doubles play: 66 and 55 are already calculated; 44 bears off (1 roll), 33 bears off, 22 bears off. Three rolls that bear off. 16+3 = 19 winning rolls.

19 is more than 18 (half of 36) so he should double and you have a very easy take.

Here are some positions for practice:

1.

is Player 2 score: 0 pip: 7 | ||||||||||||||||

Unlimited Game | ||||||||||||||||

pip: 3 score: 0 is Player 1 | ||||||||||||||||

XGID=-AA------------------aa---:1:-1:-1:00:0:0:0:0:10 | ||||||||||||||||

on roll, cube action? |

2.

is Player 2 score: 0 pip: 6 | ||||||||||||||||

Unlimited Game | ||||||||||||||||

pip: 3 score: 0 is Player 1 | ||||||||||||||||

XGID=-AA-----------------a---a-:1:-1:-1:00:0:0:0:0:10 | ||||||||||||||||

on roll, cube action? |

3.

is Player 2 score: 0 pip: 4 | ||||||||||||||||

Unlimited Game | ||||||||||||||||

pip: 3 score: 0 is Player 1 | ||||||||||||||||

XGID=-AA--------------------b--:1:-1:-1:00:0:0:0:0:10 | ||||||||||||||||

on roll, cube action? |

4.

is Player 2 score: 0 pip: 8 | ||||||||||||||||

Unlimited Game | ||||||||||||||||

pip: 3 score: 0 is Player 1 | ||||||||||||||||

XGID=-AA-----------------a-a---:1:-1:-1:00:0:0:0:0:10 | ||||||||||||||||

on roll, cube action? |

1. double, take. I count exactly 18 rolls: 22, 33, 34, 35, 36, 44, 45, 46, 55, 56, 66, and so 50% of the time he will win 2 points and 50% of the time he will lose 2 points. So it doesn't matter if he doubles or not. His opponent has a very easy take, but he /might/ not know that and might drop, so it's better to double than not.

ReplyDelete2. double take: 21 rolls: 22, 33, 44, 55, 66, 15, 25, 26, 35, 36, 45, 46, 56

3. double drop. the only bad roll here is 21.

4. no double, take: 14 rolls: 33, 44, 55, 66, 35, 36, 45, 46, 56

--pthalo

1. 17 rolls (the ones mentioned by pthalo) => no double, take.

ReplyDelete2. 23 rolls (the ones mentioned by pthalo + 61) => double, take.

3. 26 rolls (11 + all rolls without 1) => double, take.

4. 14 rolls (33+44+55+66+35+36+45+46+56) => no double, take.