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

Options

A. A. `(1)/(54)`
B. B. `(1)/(108)`
C. C. `(53)/(54)`
D. D. `(107)/(108)`

Answer

C

Explanation

`(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)`