Backgammon Statistics To Know
These were all more or less directly copied from the source attributed at the bottom of the answer:


Directly rolling a particular number (e.g. 2)    30.55%
Rolling a particular double (e.g. 3-3)   2.77%
Rolling a particular non-double (e.g. 5-1)   5.54%
Rolling any double   16.66%


Chance of getting off the bar with one or two pieces and X open points:

Open
Points  1 piece 2 pieces
1       31%     3%
2       55%     11%
3       75%     25%
4       89%     44%
5       97%     69%
6       100%    100%


Chance of moving X points in a single roll: (direct and indirect combined; direct alone is always 31% for 1-6 only).

Move    Direct and
        Indirect Chance
1       31%                      
2       33%                     
3       39%                       
4       42%                       
5       42%                           
6       47%         
7       17%
8       17%
9       14%
10      8%
11      6%
12      8%
13       -
14       -
15      3%
16      3%
17      -
18      3%
19       -
20      3%
21       -
22       -
23       -
24      3%


SOURCE (and more stats): http://www.paulspages.co.uk/bgvaults/tips/dicerolls.php
More info on observed Backgammon statistics: http://www.bkgm.com/motif/stats.html