TRUE OR FALSE : Events in a partition cannot be independent (assumption: every event in partition has nonzero probability) I need help solving this T or F question.
I got the following but unsure if correct.
Let A and B be two partitions in set X.
$$P(A \cap B) = 0$$
Let's assume A and B are independent, then:
$$P(A)P(B) = P(A \cap B) = 0$$
So this means either P(A) = 0 or P(B) = 0.
Because this violates the assumption that every event in partition has nonzero probability, does that mean this is True?

Yes. Let me formalize a bit. Consider a partition, i.e. some events $A_1, \dots, A_n$ such that $A_i \cap A_j = \emptyset$ if $i \ne j$ and $\cup_{i=1}^n A_i = \Omega$. Then, for all $i \in \{1,\dots,n\}$, for all $j \ne i$, $A_j \subset A_i^c$ (where $B^c = \Omega \setminus B$ for any event $B$). So, as you mentioned, for $i \ne j$
$$\mathbb{P}(A_i \cap A_j) = \mathbb{P}(\emptyset) = 0,$$ while, again as you mentioned $\mathbb{P}(A_i), \mathbb{P}(A_j) \ne 0$. My point is just that this should not surprise you because, in your setting $B \subset A^c$ from the very fact that $\{A,B\}$ is a partition (and in fact, if you have only two events, $B$ is precisely $A^c$.
