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Simple (!?) Backgammon Math
Posted By: Bob Koca
Date: Tuesday, 1 March 2016, at 8:14 p.m.
In Response To: Simple (!?) Backgammon Math (johnk)
The infinite sum is a valid way. The faster way is to just take the reciprocal and get an average of 36/11 rolls to enter vs 5 point board.
There are a couple ways to arrive at that (other than calculating the infinite sum). Probably the easiest to understand is to look what happens on the first try. If there is chance p of success on any given shake and N represents number of shakes needed on average then with probability p we have taken 1 turn and with probability (1p) we have used 1 turn and the situation has repeated so we have on average N more.
This gives the equation N = p (1) + (1p)(1 + N)
Solving for N gives N = 1/p.

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