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I understand that events cannot be mutually exclusive and independent (when P(event) $\neq 0$). From this, I also realized that by definition then, mutually exclusive events must also be dependent. Does it hold then, that events that are not mutually exclusive must be independent?

Any mathematical/intuitive explanation is greatly appreciated, as well as how to recognize independent vs. dependent events if this is not the case.

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    $\begingroup$ "I understand that events cannot be mutually exclusive and independent" - This is only true when both events have nonzero probability. More generally it's false. $\endgroup$ – Brian Moehring Oct 4 '20 at 5:29
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    $\begingroup$ As to your question, consider both events to be the same (with nonzero probability). Are they mutually exclusive? Are they independent? $\endgroup$ – player3236 Oct 4 '20 at 5:36
  • $\begingroup$ @BrianMoehring Does "events $A$ and $B$ are mutually exclusive" mean that $A\cap B=\emptyset$ or does it mean that $P(A\cap B)=0$? I think it should be the latter but I'm not sure, I'm not an expert. $\endgroup$ – bof Oct 4 '20 at 6:39
  • $\begingroup$ @bof In my experience, it means $A \cap B = \emptyset$, so that it is independent of the probability measure. I don't think I've seen the term in the wild, though, so I wouldn't be very bothered if someone used it to mean $P(A \cap B) =0$ (e.g. in $L^p$ spaces) $\endgroup$ – Brian Moehring Oct 4 '20 at 6:59
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Abstractly, one could argue that virtually the opposite of your statement is true, i.e. that events that are not mutually exclusive are still almost never independent.

Imagine two events $A$ and $B$ that are not mutually exclusive, such that $P(A) = 0.3 \,$ and $P(B)=0.4 \,$. Consider the Venn diagram of the two overlapping sets, and visualize moving them closer together or further apart, thus varying the size of the overlapping region $A \cap B$. It should be clear that $P(A \cap B)$ could take on any value between $0$ and $0.3$, but of that infinite set of possible values, the only one that would make $A$ and $B$ independent would be $P(A \cap B) = 0.12 \,$.

It's not difficult to use the same reasoning come up with concrete examples. Imagine that in a class of 100 high school seniors, 30 take Physics and 40 take Calculus. Almost certainly those two events will not be mutually exclusive, but unless there are exactly 12 students taking both courses, the two are not independent.

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Events that are not mutually exclusive can be dependent.

Example: drawing a King or a Heart from a deck of cards. This is not a mutually exclusive event: if you draw a King, that doesn't rule out the fact that you haven't drawn a Heart. You might've drawn a King of Hearts.

$P(K\vee ♥)=P(K)+P(♥)-P(K♥)=4/52+13/52-1/52=4/13$.

You draw a card and see that it is a King but don't see the suit yet. Even though you don't know whether the event ♥ happened, because the event K happened, the probability has changed to $P(K\vee ♥)=1$ (Technically, $P(K\vee ♥|K)=1$, the probability of "K or ♥ given K").

If knowledge of event A changes the probability of event B, the two events are dependent.

The knowledge that you drew a King changes the probability of the event $K\vee ♥$ (drawing a King or a Heart), therefore the two events are dependent.

(If you drew a card and saw that it was not a King, the probability would have changed to $P(K\vee ♥|\neg K)=3/13$; regardless of the outcome, this is still a dependent event.)

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