Compute $E(1_{W\leq t}\mid X)$

Question

Let $(X,Y)$ be independent random variables with $X>0$, and $W = XY$.

How can I compute $E(1_{W\leq t}\mid X)$ using the distribution function of $Y$?

Some help would be appreciated

Answer 1

The expectation is equivalent to $P(W\leq t \mid X)$ which is equal to $P(Y\leq t/x)$ when $X=x\neq 0$.