Probabilities: Backgammon - final move
Let's set a lower bound on the likelihood of winning with doubles, by simply ignoring all cases where it is impossible to win without doubles.

Assumption: All board positions considered are equally likely. This is probably not true, but will approach truth in longer games.

Consider the case of two men only left on the board, both in the home court, and not both on the 6-point (so that it is possible to win without doubles). There are 20 possible board positions for the two men, of weight 2 when on separate points and of weight 1 when on the same point. Likewise there are 30 possible non-double dice rolls and 6 possible double rolls. Averaged over all (the assumed equally likely) board positions under consideration, a simple counting exercise gives a probability of winning without doubles of 550 out of 1050, and a likelihood of winning with doubles of 168 / 210. (Note that the total possibilities = 1050 + 210 = 1260 = 35 * 36 as required.)

These are probabilities of 53.9% and 80% respectively, with a ratio of double-win to total wins of 23.4%.

This ignores the relatively small case-set of single blot board positions and the significantly larger case-set of board positions with 3 or 4 pieces or one or more pieces out of the home board. The net affect of these will clearly increase the likelihood of winning with a double roll.

This analysis can be extended over the additional case-sets, but it is clear that the likelihood of winning with a double roll far exceeds the 16.7% chance of rolling doubles.

                   **Wins**       **Wght * Wins**
Wght  Position  Single  Double   Single    Double
 1      1 - 1      30      6       30         6
 2      1 - 2      30      6       60        12
 2      1 - 3      28      6       56        12
 2      1 - 4      26      5       52        10
 2      1 - 5      24      5       48        10

 2      1 - 6      22      5       44        10 
 1      2 - 2      20      5       20         5
 2      2 - 3      20      5       40        10 
 2      2 - 4      18      5       36        10 
 2      2 - 5      16      5       32        10 

 2      2 - 6      14      5       28        10 
 1      3 - 3      12      5       12         5
 2      3 - 4      12      5       24        10 
 2      3 - 5      10      5       20         8
 2      3 - 6       8      4       16         8

 1      4 - 4       6      4        6         4
 2      4 - 5       6      4       12         8
 2      4 - 6       4      4        8         8
 1      5 - 5       2      4        2         4
 2      5 - 6       2      4        4         8
----------------------------------------------------------------
35                                550/1050   168/210