Given two random variables $X_1\sim\text{Exp}(\lambda_1)$, $X_2\sim\text{Exp}(\lambda_2)$ find $$\mathbb P(X_1<X_2\,\vert\,X_1>a)$$ where $a\geq0$.
Clearly one may use the memoryless property and only compute $\mathbb P(X_1<X_2)=\lambda_1/(\lambda_1+\lambda_2)$, but suppose I wanted to use the conditional definition, then $$\mathbb P(X_1<X_2\,\vert\,X_1>a)= \frac{\mathbb P(a<X_1<X_2)}{\mathbb P(\,X_1>a)}=e^{a\lambda_1}\int_a^\infty\int_{x_1}^\infty\lambda_1\lambda_2e^{-(\lambda_1x_1+\lambda_2x_2)}dx_2dx_1=\frac{\lambda_1}{\lambda_1+\lambda_2}e^{-\lambda_2a}$$ which doesn't agree with the initial answer. What is wrong?
Using the memoryless property to "cut away" everything less than $a$ ignores the possibility that $X_2<a$. If we had had $$ P(X_1<X_2\mid X_1>a, X_2>a) $$ then the memoryless approach would've worked. But that's not what we are asked for.