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

>

> quasi

Neat! (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