An application of law of total expectation - product of events The law of total expectation says
$$E(X) = E(E(X\mid Y))$$
If I am computing the expected value of the product of two events, can I say
$$E(AB) = E(E(AB\mid A))?$$

Yes.
You can further say: $$\begin{align}\mathsf E(AB)~=~&\mathsf E\big(\mathsf E(AB\mid A)\big)\\[1ex] ~=~& \mathsf E\big(A\,\mathsf E(B\mid A)\big)\end{align}$$

$\tiny\text{Small print: With usual provisos that $B$ is $\sigma(A)$ measurable and such.}$
