# Obtaining expectation of a variable given a conditional expectation

Is there any mathematical association between the conditional expectation of a variable given another variable, and the unconditional expectation of that variable?

I realise that given a joint distribution between $$A$$ and $$B$$, we can find $$E[A]$$ by first finding the marginal distribution of $$A$$.

However, suppose we know what is $$E[A|B]$$, is it possible to obtain $$E[A]$$ from this alone using any mathematical property?

No. You also need additional information to obtain $$\mathrm{E}(A)$$. In particular, the probability distribution of $$B$$ helps. We can write
$$\mathrm{E}(A) = \sum_b \mathrm{E}(A|B=b) \mathrm{P}(B=b)$$
This means that if you know $$\mathrm{E}(A|B)$$ and $$\mathrm{P}(B)$$, then you can calculate $$\mathrm{E}(A)$$.