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Topic: probability question about the dice game
Replies: 21   Last Post: Feb 18, 2013 2:47 PM

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Frederick Williams

Posts: 2,164
Registered: 10/4/10
Re: probability question about the dice game
Posted: Feb 15, 2013 8:00 AM
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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



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