Suppose we have some independent trials with outcomes $1,2$ with probabilities of $p_1$ and $p_2$ and $p_1+p_2=1$. Now suppose we have the expected number of trails until 1 occurs $n$ times $E(X_n)=a$ and expected number of trails until 2 occurs $n$ times $E(Y_n)=b$. Now if I want to find the expected number of trials until 1 or 2 occurs $n$ times can I say that it is $a+b$? I am pretty sure this is wrong but any help is appreciated. Thanks
First I think you need to work out the probability that 1 occurs n times given that 2 hasn't occurred n times (P1n, say) and vice versa.
There you can work out E(Xn | not(Yn) ) and vice versa easily enough. And the answer you are looking for should be P1n.E(Xn | not(Yn)) + P2n.E(Yn | not(Xn)).
Sorry for the poor formatting. This is my first answer. Perhaps someone can help out with that.