Understand probability: win at Backgammon.

Backgammon is played by two people who throw a pair of dice each turn, and move pieces around a board according to the throw. If a player’s piece lands on an opponent’s piece it is taken, and sent back to the start. Now, the most likely total of pips on two dice is 7 – the “possibility space diagram” (I call it a spotty diagram!) shows that there are six possible throws giving 7, more than for any other total. So is it a good tactic to avoid placing any of your pieces 7 points away from your opponent?


Total of 7 on 2 dice

No, because in a unique backgammon twist, the numbers on the two dice can either be totalled, or used separately. For example, if you throw a 4 and a 3, you can either move one piece 7, or you can move one piece 4 and another piece 3. This means that the probability of being able to move a piece 6 or fewer points increases dramatically. My second spotty diagram shows all the possible throws which would allow you to move a piece 4 points: either by getting a total of 4 (three possibilities), or if there is a 4 showing on either dice (eleven possibilities). 14 possible throws out of 36, over twice as likely as getting a 7. Even a move of one point, impossible using totals, can be achieved in 11 different ways.


Total of 4, OR a single 4

Whenever I’ve played backgammon I’ve used an instinctive feel for the probabilities to decide the safest positions for my pieces. I’d never done the calculations, so thought I would for this blog. It turns out that the most dangerous spot is 6 points away from an opponent’s piece – perhaps you could do your own spotty diagram to show there are 16 possible throws. The full table is:


Since there are 36 possible throws of two dice, the probability of being able to move a piece 6 points is \frac{{16}}{{36}} or nearly half. As soon as I’d worked them out, I was struck by the fact that the probabilities add to more than 1. Can you see why this is? A clue – are all the listed events mutually exclusive?

If I left it there, readers who are Backgammon players would soon be in touch to tell me that I’ve got the table wrong. Why? Because if you throw a double in Backgammon, the pips are counted twice over. For example, a double three counts as four threes, and you can: move one piece 12; move one piece 9 and one piece 3; move two pieces 6 each; move two pieces 3 each and one piece 6; or move four pieces 3 each. I think I’ll refine the calculations another day!

(There are plenty of websites where you can play Backgammon: this one is as good as any).

Image source: Wikipedia

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