```Date: Feb 15, 2013 8:00 AM
Author: Frederick Williams
Subject: Re: probability question about the dice game

quasi wrote:> > starwayinc 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> > Define events A,B,C by> >   A = event that Ann's higher score exceeds Bob's.> >   B = event that Bob's higher score exceeds Ann's.> >   C = event that the higher scores are equal.> > The goal is to find P(A).> > By symmetry, P(B) = P(A), hence> >   P(A) = (1 - P(C))/2> > When rolling 2 dice, let p_1, p_2, ..., p_6 be the respective> probabilities that higher score is 1,2, ... 6.> > Then> >    P(C) = (p_1)^2 + (p_2)^2 + ... + (p_6)^2> > Noting that> >    p_1 = 1/36>    p_2 = 3/36>    p_3 = 5/36>    p_4 = 7/36>    p_5 = 9/36>    p_6 = 11/36> > we get> >    P(C) = (1/36)^2 + (3/36)^2 + (5/36)^2 + ... + (11/36)^2> >         = (1^2 + 3^2 + 5^2 + ... + 11^2)/36^2> >         = 286/1296> > hence> >   P(A) = (1 - (286/1296))/2> >        = 505/1296> > quasiNeat!  (Or 'cool' as the modern young people say.)-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him.Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
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