Finding 01 or 00 A and B play a bit game. Unbiased bit generator is generating 0 or 1 repeatedly until one of the following happens.


*

*The bit patterns to '00' (i.e., a 0 is immediately followed by a 0) for the first time. In this case A wins.

*The bit patterns to  '01' (i.e., a 0 is immediately followed by a 1) for the first time. In this case B wins.
Who has more probability of winning the game?
I am getting both probability to be the same. Can anyone help me out how to approach this question?

In this scenario, note that once the first $0$ is generated, then the next number generated determines the victor of the game. The only thing that could happen prior to the first $0$ is some finite sequence of $1$'s which wouldn't help anyone, so we don't care about anything until that first $0$.
In which case the conclusion is clear - there's a $1/2$ probability to get a $0$ on the succeeding digit, and $1/2$ for a $1$.
Thus, the probabilities of each winning are the same.
