Question

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

A and B cannot be disjoint
A and B must be independent
A and B cannot be independent
A and B must be disjoint
It is always true

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)`