Conditional probability of complementary events

Question

Let $X$ be a random variable, and consider two events $E$ s.t. $p(E) > 0$. Denote with $E^c$ the complement of $E$, and suppose that $p(E^c) > 0$. Then for any event $E'$ with $p(E') > 0$ it holds $$ \mathbb{E} [X \mid E'] = p(E) \mathbb{E} [X \mid E', E] + p(E^c) \mathbb{E} [X \mid E', E^c]. $$

Answer 1

The statement is not correct.

What we do have under suitable conditions is:$$\mathbb{E}\left[X\mid E'\right]=P\left(E\mid E'\right)\mathbb{E}\left[X\mid E',E\right]+P\left(E^{\complement}\mid E'\right)\mathbb{E}\left[X\mid E',E^{\complement}\right]$$