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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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quasi

Posts: 10,451
Registered: 7/15/05
Re: probability question about the dice game
Posted: Feb 14, 2013 12:11 PM
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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



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