Question

What must be true in order for p (a or b) = p(a) + (b)

Answer 1

A and B must be disjoint. You have, by the inclusion/exclusion principle, that

$\mathbb P(A\cup B)=\mathbb P(A)+\mathbb P(B)-\mathbb P(A\cap B)$

where the last term is required to prevent double counting. When $A$ and $B$ are disjoint, you have $A\cap B=\emptyset$ and $\mathb P(A\cap B)=0$, leaving you with

$\mathbb P(A\cup B)=\mathbb P(A)+\mathbb P(B)$