Count the 36 ordered outcomes
There are 36 equally likely ordered two-die outcomes. The rolls 4–1 and 1–4 are the same to play, but each has its own outcome. Each non-double therefore counts twice and each double counts once. The chance of rolling at least one specific number is 11/36: six outcomes have it on the first die, six on the second, and one appears in both lists.
A direct shot is a hit achievable with one die. An indirect shot needs multiple dice. To count an indirect shot, check every useful dice combination, the intermediate landings, and any bar priority or maximum-dice rule. A point can be seven or more pips away and still be hittable in one turn.
Umber can hit point 11 from 7 with a direct 4. With 3–1, Ivory blocks both intermediate landings at 8 and 10.
Duplicate an opponent’s useful numbers
Duplication means two things the opponent wants to do call for the same number. Giving the opponent a direct 4 to hit two separate targets can be safer than giving them a direct 4 at one target and a direct 5 at another. If every 4 hits at least one target and there are no other hits, its 11 out of 36 outcomes are shared. At least one 4 or 5 appears in 20 out of 36 outcomes.
The comparison is conditional. A double can sometimes do both jobs. Alternate routes may add hits, and after the first hit the best second move can change. The number of rolls that hit matters alongside what the position looks like if they hit once, hit twice or miss entirely.
Umber has a direct 4 from 7 to hit on 11, and another direct 4 from 9 to hit on 13. One ordinary 4 cannot make both hits. Count the distinct roll outcomes as well as the two destinations.
Account for return shots after you hit
When you hit, look at the checker you leave on the hitting point. If the hit was in your home board and the opponent is on the bar, the die that matches the point may hit straight back. If you can cover with the other die, the opponent loses that entry number. If your own home board is weak and theirs is strong, a return hit can cost far more than you gained from the first hit.
Try comparing two candidate plays by listing the opponent’s replies in three groups: hits, strong improvements without a hit and awkward rolls. This gives more information than counting blots alone. Even when the total number of hits is equal, one blot may be much more expensive to lose.
Ivory can play 6/3* and 8/7 with 3–1, leaving a lone checker on 3. A 3 on Umber’s next roll hits that blot on entry if the rest of the turn allows it.
Practice question
All rolls containing a 3 or a 6 hit a blot. There are no other hits. How many of the 36 outcomes hit?
Show answer
20. There are 11 outcomes containing each number. Subtract the two orders of 3–6, which both counts included.