A die is thrown three times events A and B are defined as below: `A=` Getting 4 on third die, `B=` Getting 6 on the first and 5 on the second throw Find the probability of A given that B has already occurred.

Explanation

The sample space has 216 outcomes.

Now `A=(1,1,4)(1,2,4) ldots(1,6,4)(2,1,4)(2,2,4) ldots(2,6,4)(3,1,4)(3,2,4) ldots(3,6,4)` `(4,1,4)(4,2,4) ldots(4,6,4)(5,1,4)(5,2,4) ldots(5,6,4)(6,1,4)(6,2,4) ldots(6,6,4)`

`B={(6,5,1),(6,5,2),(6,5,3),(6,5,4),(6,5,5),(6,5,6)}`

And, `A cap B={(6,5,4)}`.

Now, `P(B)=6 / 216` and `P(A cap B)=1 / 216`

Then `P(A / B)=(P(A cap B)) / P(B)=(1 / 216) /(6 / 216)=1 / 6`