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In a game of chance a player throws a pa...

In a game of chance a player throws a pair of dice and scores points equal to the difference between the numbers on the two dice. Winner is the person who scores exactly `5` points more than his opponent. If two players are playing this game only one time, then the probability that neither of them wins to

A

`(1)/(54)`

B

`(1)/(108)`

C

`(53)/(54)`

D

`(107)/(108)`

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The correct Answer is:
C
`(c )` Player `A` can win if `A` throws `(1,6)` or `(6,1)` and `B` throws `((1,1),(2,2),(3,3),(4,4),(5,5) or (6,6))`. Thus the number of ways is `12`.
Similarly the number of ways in which `B` can win is `12`.
Total number of ways in which either `A` wins or `B` win `=24`
Thus the number of ways in which none of the two wins `=6^(4)-24`.
`:.` The required probability `=(6^(4)-24)/(6^(4))=(53)/(54)`
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