Two archers take alternating turns trying to hit the opposing archer. As soon as the shooting archer hits the other, the shooting archer wins.
Both have a specific probability p to hit the other.
p1 : Probability of the archer who starts p2 : Probability of the second archer
Now what is the probability that the archer who starts wins for any p1 and p2?
The starting archer has a big advantage, because he can shoot first, so he has atleast his p1 probability to win outright.
Probability for first archer to win = p1+x
But then it could go on infinitely without anyone hitting the other, how can that x be described/calculated?
Let $P_1$ be the probability of the first archer winning. The trick is to notice that if the first two shots both miss, then the situation is the same as it was at the start, so the first player's probability of winning is $P_1$ again. So we have:
$$P_1=p_1+(1-p_1)(1-p_2)P_1$$
You can easily solve this for $P_1$.