Two soccer players compete in a penalty shoot-out. Player A scores every other shot, and player B scores every third shot. They take turns shooting until one of them scores a goal.
What is the probability that player B will win, if he goes first?
The probability that B will win is 50%. You can see this in the following diagram. There are three possibilities per round:
Every result has a probability of 1/3. If the match ends in a standoff, it starts over with the same probabilities for the three possible results.