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

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

  2. 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?

Nishant Raj
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    The probabilities are indeed the same. Nothing really happens until the first $0$ and then the next bit determines the winner with equal probabilities – Henry Mar 20 '19 at 08:44

2 Answers2

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

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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.

PrincessEev
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