donno how to get those pics (and no idea if I'm allowed to post it), so I'll describe the situation. Black has 2 men on 23-point directly in front of white's 5-prime, 12 men close his entire inner board, and the last one on his 8-point. White has 2pieces on the bar. Black holds the cube.

The problems asks if Black should double and if white should take, and the answer is double and pass.

I wonder if this should be played for gammon cos black has 11/36 to jump out on the roll, and 11/36 * 15/36 on next roll and still maintaining his board. So that's 43.3%, among those 43.3%, I guess 85% will lead to a gammon and 10% will be a single win and 5% lose. This will generate 1.61 EV while double and pass has EV=2. To make it 2, black will need to win about 41% if he cannot jump out before his board breaks.

So is it true that black has too little equity when his board breaks to make play for double a bad idea?

Now if this is true, white can take knowing that if black fails to do those 43.3, he can immediately redouble.

Ok, your assumptions about the winning chances when jumping the prime seem reasonable, so let's go with them.

43.3% of the time we jump the prime, and, given your assumptions, we'll then have an EV of 0.85 * 4 + 0.10 * 2 - 0.05 * 2 = 3.5

We would be indifferent to doubling if our EV x when playing on and not jumping the prime brings the overall EV to 2.

0.433 * 3.5 + 0.567 * x = 2
=> x = 0.855

So our equity must be 0.855 in case we don't jump the prime to make us indifferent. If it is better, we play on for a gammon. If it is less, we double to take the 2 points down.

A GNU rollout says that if we play on, we'll win 61% with 48% being gammon wins. We have already covered 43.3% x 85% = 36.8% of the gammon wins with jumping the prime. This leaves 11.2% gammons if we don't jump. We also have covered 43.3% x 10% = 4.33% of the normal wins if we jump.

Of our losses, we have already accounted for 43.3% x 5% = 2.2% when we jump the prime. This leaves us 36.8% losses if we don't jump. GNU says we'll lose a gammon 8.6% of the time and I assume strongly that all of them happen if we don't jump.

So if we don't jump, we win 61% - 36.8% - 4.33% = 19.87% with 11.2% being gammons. We lose 28.2% normal games and 8.6% gammons.

Our EV here is 0.0767 * 2 + 0.112 * 4 - 2 * 0.282 - 4 * 0.086 = -0.307

So overall we have: 0.433 * 3.5 + 0.567 * (-0.307) = 1.34

This result is roughly equal to what GNU says (note that the GNU result is normalized to 1 point, and that the above analysis doesn't take backgammon wins into account)

And a caveat - I suck at pen and paper maths... so there might well be errors in my analysis...

Good job. Your analysis looks fine.

It's worth noting that this position is one of a class of positions that the bots evaluate very poorly. A Snowie level 3 evaluation rates this as no double, take with an after-doubling equity of -0.6 (that's equity for the taker, on a cube going from 1 to 2). Its rollout, however, then judges it as double, pass with an after-doubling equity of -1.06. That margin of error (0.40) is huge for a bot and just illustrates how difficult and razor-edged these positions really are.