Probability of two exponential random variables take the same value

Question

Let $A$ and $B$ be independent random variables drawn from the exponential distribution with parameters $\lambda_A$ and $\lambda_B$. What is the probability that $A=B$?

Answer 1

If $X$ and $Y$ are independent continuous random variables, then $$ \mathsf{P}(X=Y)=\mathsf{E}[\mathsf{P}(X=Y\mid Y)]=\int \mathsf{P}(X=y)f_Y(y)\, dy=\int 0 f_Y(y)\, dy=0, $$ where $f_Y$ is the pdf of $Y$.