
Re: probability question about the dice game
Posted:
Feb 14, 2013 12:47 PM


On Thursday, February 14, 2013 9:31:48 AM UTC8, Jussi Piitulainen wrote: > Ray Vickson writes: > > > > > On Thursday, February 14, 2013 5:29:06 AM UTC8, starw...@gmail.com 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 singletoss 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. > > > > Your final denominator is 6^8 for a problem involving four tosses. > > I'd've expected 6^4. Should the denominator be only 6^2 for p[i]? > > > > And surely p[i] should be defined on 1, ..., 6, not on 2, ..., 12. > > The latter looks like the probability of a sum instead of a max. > > > > > Let X = score of A and Y = score of B. The momentgenerating > > > 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 = XY 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.
A single toss of TWO dice has denominator 36, so two tosses of two dice has denominator 36^2, and the difference between the two maxima has denominator 36^4. However, as stated in my previous post, I thought the question involved two tosse of 2 dice each, rather than two tosses of a single die. That mixup arose from language usage.

