An exam question asked to select all expressions that are equal to 1, given no independence assumptions. The solution stated that the following expression was not equal to 1, but I don't understand why. Is it possible this is a mistake?
$\sum_{a}\sum_{b}P(A=a | B=b)$
For a fixed $b$, the sum is $1$ since you go over all possibilities of $A$. But then you also sum over all values of $b$, which means you sum over many $1$s.
For example, suppose you throw a dice. $A$ is the value of it, $B=1$ is it is even, $0$ if odd. Then this sum should be $2$, as given even, the sum of all options of $A$ is $1$ and given odd the sum of all options of $A$ is $1$.