hi folks, i'm reading "501" right now. mr. robertie explains the beaver concept. this i understand, but i'm not sure about the math. in his example, white has one checker on the 22pt and one checker on the 20pt. black has two checkers on the 2pt. if I understand this correctly, white has a 39% chance of winning. now do I calculate black's chance as 61% times 72% which equals 44% ? and if this is right, why should black beaver if he's an under dog? what am i missing? thanks in advance for any help.

great book and a great site mr. robertie

[img]http://s9.************/7z18wsefz/Screenshot_from_2013_12_24_11_49_49.png[/img]
screen shot pc

You are making two mistakes. First, you are thinking in terms of probability rather than equity, and second, you need to factor in Black’s recube.

Equity, or expected net winnings, is the difference between what you win and what you lose. Equity is negative when you lose.

Assuming a centered cube, we start with White doubling to 2, and Black taking. Then we calculate White's equity. If White’s equity is negative, then Black should beaver.

After he doubles, White will win immediately with 14 numbers (66, 55, 44, 33, 65, 64, 63, 54, 53). Since the cube will be at 2, the contribution to White’s equity is 2 * (14/36).

On the other 22 rolls, Black will get a turn. This shows up as the (22/36) in the calculations below.

When Black gets a turn, he will redouble to 4, and White will take. Black then wins immediately with 26 numbers and loses with the other 10. He loses when he rolls any non-doublet ace (61, 51, 41, 31, 21). Note that Black will lose even if White rolled 21 on his turn. That’s because White can give a redouble that Black must pass. On the 10 rolls when White wins, the contribution to his equity is 4 * (22/36) * (10/36). On the 26 rolls when White loses, the contribution to his equity is –4 * (22/36) * (26/36). The sum of these is –4 * (22/36) * (16/36).

White’s total equity is thus:

Equity( White ) = 2 * (14/36) – 4 * (22/36) * (16/36)

= (2 * 14 * 36 – 4 * 22 * 16 ) / (36 * 36)

= –400/1296

After the beaver, White’s equity drops to –800/1296. Of course, White could pass the beaver, but if he does, his equity plummets to –2.

Mike

In the explanations above, change 'Black' to 'White' and vice-versa and it will match the posted disgram.

That is true as far as the colors in the diagram are concerned. But note that Yogiman has placed the checkers in the wrong home boards. Black's checkers are in White's home board, and vice versa.

I used the terms "Black" and "White" in the same sense as the original poster.

Mike

If White doubles, should Black beaver?

These are the colors used in the original post.

Mike

Thanks to Robertie and Mike.

Apologizing is for weaklings.

[img]http://s18.************/p6u6iphfd/Screenshot_from_2013_12_25_07_36_29.png[/img]
screen shot windows 7

PS Still apologizing, because it is Christmas. By the way, this is for a 5-match

thanks Mike, it never occurred to me to think in terms of equity. this makes sense to me now. I appreciate the help.