#### Question

A and B throw a die alternatively till one of them gets a number greater than four and wins the game. If A starts the game, what is the probability of B winning?

#### Solution

Let S denote the success, i.e. getting a number greater than four and F denote the failure, i.e. getting a number less than four.

`∴ P(S)=2/6=1/3, P(F)=1−1/3=2/3`

Now, B gets the second throw, if A fails in the first throw.

∴ P(B wins in the second throw) =` P(FS) = P(F)P(S) = 2/3xx1/3`

Similarly, P(B wins in the fourth throw) = P(FFFS) = P(F)P(F)P(F)P(S) = ` (2/3)^3xx1/3`

P(B wins in the sixth throw) = P(FFFFFS) = P(F)P(F)P(F)P(F)P(F)P(S) `= (2/3)^5xx1/3` and so on.

Hence,

P(B wins) = `2/3xx1/3 + (2/3)3xx1/3 + (23)5xx1/3 + ...`

`=2/3xx1/3xx[1+(2/3)2+(2/3)4+......`

`= 2/3xx1/3xx(1/(1−4/9)) ` [∵a+ar+ar^{2}+... =a/(1−r)]

`= 2/5`

Thus, the probability that B wins is 2/5.