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?
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.
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 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