Suppose that $ X,Y$ are independent unit $\text{Exp}(1)$ rv's find $P(X<3|X+Y) $.
I am hoping to solve this problem, but I am actually not sure how to set up things correctly. I think we need to find the Density of $Z=X+Y$ and we can do something with that, but I am uncertain.
Any advice would be appreciated.
Tip: The sum of two independent exponential distributions has this pdf.
$$\begin{align}f_{\small X+Y}(z) &=\frac{\mathrm d~~}{\mathrm d z}\int_0^z\int_0^{z-x}\mathrm e^{-(x+y)}\mathrm d y\mathrm dx \\[1ex]&= z\mathrm e^{-z}\mathbf 1_{0\leqslant z}\end{align}$$
Which leads to the conditional distribution$$f_{\small X\mid X+Y}(x\mid z) = \dfrac{f_{\small X}(x)\,f_{\small Y}(z-x)}{f_{\small X+Y}(z)}\\\vdots$$