From The Backgammon Book, by Oswald Jacoby and John R. Crawford
double to 4?
It is your roll, and there are nineteen out of thirty-six rolls that will end the game immediately (double 6, double 5, double 4, double 3, double 2, 6-5, 6-4, 6-3, 6-2, 5-4, 5-3, 5-2). If black’s two men were on his one point, you should double or redouble, and thereby double your plus expectation. But here is black with a poorer position than that, and this time you should not double if the cube is on your side. The reason is that if you keep the cube on your side and fail to get both your men off, you may get a second turn. Black has seven possible rolls that won’t take both his men off.
Thus if you don’t double, your chance of winning is 19/36 plus the product of 17/36 (your chance of not getting off in one roll) times 7/36 (black’s chance of not getting off). This represents a total chance of winning of 803/1296 (19/36, or 684/1296, plus 17/36 times 7/36, or 119/1296); your chance of not winning is 1296 less 803, or 493 out of 1296. Thus your net plus expectation is 803/1296 minus 493/1296, which is 310/1296; to express it as a decimal, divide 310 by 1296, which comes to .239 times the stake.
If you do double, your opponent will accept, and now your whole chance depends on your first roll, because if you don’t get off you are going to be redoubled, and you will have to refuse. Your net plus expectation if you double is therefore 2/36 (19/36 less 17/36) times 2 (the doubled stake), which is 4/36 or .111 times the undoubled stake.
In other words, you do better than twice as well if you leave the cube alone.