Expected value mixed poisson process

Question

I have a geometric counting process defined as a mixture between the family of homogeneous Poisson process with respect to the random intensity $\Lambda$ which is exponentially distributed with parameter $\mu=2$. I have some issues with the request to find the $E[N(1)|N(2)=1]$. I tried with the fact that $E[N(t)]=E[\Lambda]t$ but I'm stuck because I don't know how to manage the condition $N(2)=1$.
Could be that $N(1)|N(2)=1 $ is $Bin(1,1/2)$ and for this reason the $E[Bin(1,1/2)]= 1/2 $ Thanks for your help