White doubles. Should Black take?

The race says no. The possible chance to hit says yes. How to calculate which reasoning wins?

I would figure that I hit 15% of the time next roll, which is a number I have memorized, but you could also count up the blot numbers for White and figure you hit a third of those. And since I win a large majority of the hits, I don't need to win a ton games where I don't hit next roll to get to 25%. So it's pretty clear to me the race is close enough to easily get there.

Also, there's a lot of pip "wastage" in White's inner board structure, so the race might even be closer.

Seems like a snap take.

Right, I wasn’t even looking at the exact pip count. I just noted the outfield checkers are dead even and the White’s infield checkers are only a bit better than Black’s, So maybe it’s close to a take on the race alone. Add in the hits and one should be able to be very confident this isn’t close.

Can you say how you get this number 15 %

You can see my picture, i get a 19% (but probably its not the same probability wanted).

https://imgur.com/a/vNbruxU

It’s a different probability. The 19% is black’s win probability if he takes. The 15% was an estimate of black’s probability of hitting a blot. Black doesn’t automatically win when hitting, nor does he automatically lose when missing, so the two are not the same.

By my count, black has an opportunity to hit on the following rolls by white: 66, 51, 42, 41, 33, 32, 31, 22, 21, and 11. That is 16/36 probability of even getting a shot. Black will actually hit with probability 11/16 after all of these except 21, in which case he hits with any 3 plus 11 or 21 - 14/36 probability. His probability of hitting is then 2/36 x 14/36 + 14/36 x 11/36 = 182/1296 or 14% - pretty close to the 15% estimate given.

However based on the 19% win chance it looks like it is a pass.

EDIT - I forgot that black would have a 12/36 chance after white rolls 22 or 31 and that he’d have a 14/36 hitting chance after 33 as well. Revised probability

3/36x12/36 + 3/36x14/36 + 11/36x11/36 = 199/1296 = 15.4%

I believe still small error. I think you meant 3/36x12/36 + 3/36x14/36 + 10/36x11/36 = 190/1296 or ~14.66%. Right?

Yes but even that isn’t quite it. I left out 11 and 42 as rolls that leave a blot 2 pips away, and I forgot about 66 keeping it where it is, so now I’m going to be more careful about it

Blot 6 away - 66. Hit probability 17/36 (any 6 plus 51,42,33 or22)
Blot 3 away - 33, 21. Hit probability 14/36 (any 3 plus 21 or 11)
Blot 2 away - 42, 31, 22, 11 Hit probability 12/36 (any 2 plus 11)
Blot 1 away - 41, 32, 51 Hit probability 11/36

1/36x17/36 + 3/36x14/36 + 6/36x12/36 + 6/36x11/36 = 197/1296 = 15.2%

Of course for all practical purposes my original calculation and the estimate of 15% would be perfectly fine. I just hate when I mess up things like this and wanted to make sure I got it right.

I don't know where the 19% is coming, but I have a feeling it's higher than this.

I don’t vouch for the accuracy, but 19% was the win probability on an analysis that was posted.

I appreciate your not only your dedication to precision, but also the style and prose you write with. Thanks for all your posts!