A fair die is tossed repeatedly. `A` wins if it is `1` or `2` on two consecutive tosses and `B` wins if it is `3,4,5` or `6` on two consecutive tosses. The probability that `A` wins if the die is tossed indefinitely is

Options

A. `(1)/(3)`
B. `(1)/(4)`
C. `(5)/(21)`
D. `(2)/(5)`

Answer

C

Explanation

`(c )` Let event `S to A` is getting `1` or `2`
Event `F to B` is getting `3,4,5,6`
`P(S)=P(1 or 2)=1//3`
`P(F)=P(3 or 4 or 5 or 6)=2//3`
`P(A"wins")=P[(SS or SFSS or SFSFSS or ….) or (FSS or FSFSS or….)]`
`=((1)/(9))/(1-(2)/(9))+((2)/(27))/(1-(2)/(9))`
`=(1)/(9)xx(9)/(7)+(2)/(27)xx(9)/(7)`
`=(5)/(21)`
`P(A"winning")=(5)/(21)`