Rock Paper Scissors

The Sample Space is the Cartesian Product of {Rock, Paper} and {Paper, Scissors}

Which means the following are possibilities:

(The following are ordered pair of (Bob's Move, Park's Move))

1. (Rock, Paper) -> Bob looses
2. (Rock, Scissors) -> Bob wins
3. (Paper, Paper) -> ...
4. (Paper, Scissors) -> Bob looses

Now, note that the problem says that the game continues till someone wins .

So, the (Paper, Paper) possibility is eliminated leaving:

1. (Rock, Paper) -> Bob looses
2. (Rock, Scissors) -> Bob wins
3. (Paper, Scissors) -> Bob looses

Out of this three cases only one can be won by Bob.

Clearly the chances of him winning is 1/3

The first round, there is a 1/4 chance of a win and 1/4 chance of a tie. If there is a tie, the game repeats with the same probabilities. This leads to a total probability of P(Bob wins) = P(Bob wins in first round) + P(Tie in first round) * P(Bob wins in second round) + P(Tie in first round) * P(Tie in second round) * P(Bob wins in third round) + ... .

This results in the sum (1/4)^n, of n from 1 to infinity, which is equal to 1/3.