Sum of Probabilities used to calculate Expectation value?

Question

Does the sum of all probabilities used to calculate the expectation value sum up to 1? Is it necessary OR there are some exceptions, e.g., in the case of calculating conditional expectation values?

Answer 1

Preassuming that we are dealing with expectation of a discrete random variable here (you use the words "sum up") I would say that they indeed always sum up to $1$.

If you are calculating something like $\mathbb E[X\mid X\in A]$ then you get: $$E[X\mid X\in A]=\sum_{x\in A} xp'_x$$ where for $x\in A$: $$p'_x:=P(X=x\mid X\in A)=\frac {P(X=x)}{P(X\in A)}=\frac{p_x}{\sum_{x\in A}p_x}$$

so that: $$\sum_{x\in A} p'_x=1$$