A coin is tossed. If head appears a fair die is thrown three times otherwise a biased die with probability of obtaining an even number twice as that of an odd number is thrown three times. If `(n_(1),n_(2),n_(3))` is an outcome, `(1 le n_(1) le6)` and is found to satisfy the equation `i^(n_(1))+i^(n_(2))+i^(n_(3))=1`, , then the probability that a fair die was thrown is (where `i=sqrt(-1))`

Options

A. `(1)/(12)`
B. `(1)/(3)`
C. `(27)/(59)`
D. none of these

Answer

C

Explanation

`(c )` `E:` Event of getting on outcome `(n_(1), n_(2),n_(3))` such that `i^(n_(1))+i^(n_(2))+i^(n_(3))=1`
`E_(1) : ` Event that fair die is thrown
`E_(2) :` Event that biased die is thrown
Fav. Cases :`(n_(1),n_(2),n_(3))` must be an arrangment of `{2,4,4}` or `{6,4,4}` or `{1,2,3}` , `{3,5,4}`
`P(E//E_(1))="^(3)C_(2)((1)/(6))^(3)+^(3)C_(2)((1)/(6))^(3)+3!((1)/(6))^(3)+3!((1)/(6))^(3)`
`P(E//E_(2))="^(3)C_(2)((2)/(9))^(3)+^(3)C_(2)((2)/(9))^(3)+3!((1)/(9))^(2)((2)/(9))+3!((1)/(9))^(2)((2)/(9))`
`P(E_(1)//E)=((1)/(2)[(1)/(12)])/((1)/(2)[((1)/(12))+(8)/(81)])=(27)/(59)`