Question

P(A) = 1/2, P(B) = 1/3, P(A or B) = 2/3. Are A and B mutually exclusive?

Answer 1

$A$ and $B$ are mutually exclusive if

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

You have

$\dfrac12+\dfrac13=\dfrac56\neq\dfrac23$

so the answer is no.