I'm trying to get a good grasp on the relation between these two terms. so far with no luck.
Let $(\mathbb{P},\mathcal{F},\Omega)$ be a probability space (I hope I got the notations right).
Let $A,B\subseteq \mathcal{F}$ be two non-empty events.
Let's say $A$ and $B$ are mutually exclusive, meaning $A\cap B=\phi$.
Now, on one hand, we got: $\mathbb{P}(A\cap B)=\mathbb{P}(\phi )=0$.
On the other hand; $\mathbb{P}(A)\neq 0$, and $\mathbb{P}(B)\neq 0$,
considering the last two lines, we have that: $\mathbb{P}(A\cap B)\neq \mathbb{P}(A) \mathbb{P}(B)$.
So, can I conclude that if two events are mutually exclusive that implies that they are dependent?
If the answer is yes, that doesn't make sense to me, since that will also imply that: $A,B$ are independent $\Rightarrow$ $A\cap B\neq \phi$.
Best Answer
That is (almost) true.
For two events being independent, the probability of either of them must not change given the other has occurred. If the events are mutually exclusive, the probability of one of them given that the other occurred is $0$ - and thus two events can be both mutually exclusive and independent only if both have zero probability.