Bingo ! three hit it on the head. He must be an arithmetician !
As a serious backgammon cash and tournament player (since 11/69) I needed to teach myself to think [fractionally] in terms of 1296th's as 1296 is the set of the possibilities of 2 consecutive rolls of the dice.
e.g. If the only way that I could lose a game in a particular position would be for me to fail to roll a total of 7 pips or less AND for my opponent to roll ANY double number (1-1, 2-2, etc.) then I would immediately know that I would lose if I rolled 6-1, 5-1, 4-1, 3-1, 2-1, 1-1, 5-2, 5-1, 4-3, 4-2, 4-1, 3-2, 3-1, 2-1, or 1-1 = 28/1296* AND my opponent rolls one of the 6 sets of doubles I would have 28 + 4 = 32 x 1296/100 chances of losing. "Over the table" (mental computing only permitted, as in BJ) I know that if 1296 = 100% then 1% = 12.96 combinations. In this case I have just 28 losing combo's Rounding 12.96 to 13 as representative of 1% I know that I have barely a 2% chance of a loss with 26 losing combos.
*regular rolls of the dice appear 2 ways each (e.g. 6-5 OR 5-6) while doubles appear but once.
Incidentally, the mean roll in backgammon, where a roll of doubles count as 4 times the number on one of the dice = 8.16 pips.
Mentally computing by a rounded 8 pips is also a requisite skill to play backgammon for significant money.
Backgammon is the favorite game of actuaries, statisticians, mathematicians, for a very good reason. It is a game of applied probabilities; that neither human being nor computer has completely mastered, unlike simple games like chess or bridge or go.
It would be nice if I could "think ahead" by 3 or 4 rolls but those products are 46,656 and 1,679,616; and one can easily see what happens when the combatants toss a PAIR of dice 50 to 100+ times during a single game and repeat that many many times during the course of a backgammon tournament match or a session of individual"cash" games.
