# Finding the probability of occurrence of one event before an another

In a home work problem $E$ and $F$ are mutually exclusive events in the sample space of an experiment. The experiment is repeated until either event $E$ or event $F$ occurs. Show that the probability that event $E$ occurs before event $F$ is given by $$\frac{P(E)}{P(E) + P(F)}.$$

I am assuming that this is a geometric processes with prob. that $E$ occurring is $p$ and $F$ occurring is $q$. Then $$P(E) = (1-(p + q))^{n-1} p$$ $$P(F) = (1 - (p + q))^{N-1}q$$ where $n$ it the number of trials for event $E$ to occur and $N$ is number of trials for $F$ to occur.

Is this approach correct. I am not able to proceed from here. Can this processes be anything other than a geometric processes.

The main question is: what is the probability that $E$ occurs under the condition that one of the events occur?
Something like: $$P\left(E|E\cup F\right)=\frac{P\left(E\cap\left(E\cup F\right)\right)}{P\left(E\cup F\right)}=\frac{P\left(E\right)}{P\left(E\right)+P\left(F\right)}$$