$f_{\mathbb{X},\mathbb{Y}}(x,y)=e^{-x}$ if $0\leq y \leq x < \infty$
X need to find the $P(X<3|Y<2$) and $P(X<3|Y=2)$.
I'm struggling with the first probability. I'm not sure how to evaluate the conditional given that $Y<2$. I know conditional = joint/(marginal of given) but I can't figure out how to evaluate it.
So you know to apply the definition of condiional probability. That is:$$\mathsf P(X\leqslant x\mid Y\leqslant y) = \dfrac{\mathsf P(Y\leqslant y, X\leqslant x)}{\mathsf P(Y\leqslant y)}$$
and similarly that $$\mathsf P(X\leqslant x\mid Y = y) = \dfrac{\tfrac{\mathrm d }{\mathrm d y}\mathsf P(Y \leqslant y, X\leqslant x)}{\tfrac{\mathrm d }{\mathrm d y}\mathsf P(Y \leqslant y)}$$
Now, when $\bf y \leqslant x$, $$\begin{split}\mathsf P(Y\leqslant y, X\leqslant x)&=\int_{0}^{y}\int_t^{x}e^{-s}\,\mathrm d s\,\mathrm d t\\[2ex] \tfrac{\mathrm d }{\mathrm d y}\mathsf P(Y \leqslant y, X\leqslant x) &= \int_y^x e^{-s}\mathrm d s\end{split}$$