Date: Feb 14, 2013 11:50 AM
Subject: Re: probability question about the dice game

On Thursday, February 14, 2013 5:29:06 AM UTC-8, wrote:
> two players Ann and Bob roll the dice. each rolls twice, Ann wins if her higher score of the two rolls is higher than Bobs, other wise Bob wins. please give the analyse about what is the probability that Ann will win the game

P{A wins} = 723893/1679616 =approx= .4309872018.

This is obtained as follows (using the computer algebra system Maple). First, get the probability mass function (pmf) of the max of two independent tosses, which you can do by first getting its cumulative distribution = product of the two single-toss cumulative distributions. Then get the mass function by differencing the cumulative. The pmf is p[i] = [1, 8, 27, 64, 125, 216, 235, 224, 189, 136, 71]/36^2 on i = 2,...,12.

Let X = score of A and Y = score of B. The moment-generating function (mgf) of X is MX(z) = sum{p[i]*z^i,i=2..12}, while the mgf of (-Y) is MY(z) = MX(1/z). The mgf of the difference D = X-Y is MD(z) = MX(z)*MY(z). Expanding this out we have P{D = k} = coefficient of z^k, for k = -10,...,10, and the probability that A wins is the sum of the coefficients for k >= 1.