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?

The first $0$ is followed by a $0$ or a $1$ equiprobably and the game is over.
