X,Y both have poisson distributtion with parameters $\lambda$ $\nu$ accordingly: I have to calculate $$P(X>0|X+Y)$$, my first question is, does this mean I have to calculate it for every $j$ where $X+Y=j$ , because I am not sure if I understand this correctly.
If I am, $$\frac{P(X+Y=j\space\cap X>0)}{P(X+Y=j)}=1-\frac{P(X+Y=j\space\cap X=0)}{P(X+Y=j)}=1-\frac{\nu^{j}}{(\nu+\lambda)^{j}}e^{\lambda}$$ , but something must be wrong with this calculations, since when j=0, the probability should equal to zero, but it is not.
You lost $\mathbb P(X=0)$ in last equality: $$ 1-\frac{\mathbb P(X+Y=j, X=0)}{\mathbb P(X+Y=j)}=1-\frac{\mathbb P(Y=j)\cdot\mathbb P(X=0)}{\mathbb P(X+Y=j)} = 1-\frac{\nu^{j}}{(\nu+\lambda)^{j}}. $$ This probability is equal to zero when $j=0$.