Variance as a sum of conditional variances So, for a random variable $X$ and an event $A$, the following expression of the mean value of $X$ as a sum of conditional expectations is valid, or at least, i could prove it for discrete random variables:

$$ E(X) = E(X|_A) P(A) + E(X|_{A^c})(1-P(A)) $$

My question is, is there a similar expression for variances? I remember my teacher using such a thing once, something like:

$$ Var(X) = Var(X|_A) P(A) + Var(X|_{A^c})(1-P(A)) + \ ... $$

but i forgot it and i can't find it anywhere. Even the one above, i haven't found it anywhere i looked.
If you mean total probability theorem, using the law of total variance, we have:
$$\mathrm{Var(X)} = \mathrm{E}[\mathrm{Var}(X|A)] + \mathrm{Var}(\mathrm{E}[X|A]) .$$ It is somehow well-known equation, therefore, I don't include the proof. Hope that helps!
If you mean total probability theorem, using the law of total variance, we have:
$$\mathrm{Var(X)} = \mathrm{E}[\mathrm{Var}(X|A)] + \mathrm{Var}(\mathrm{E}[X|A]) .$$ It is somehow well-known equation, therefore, I don't include the proof. Hope that helps!