If two events are not mutually exclusive, does that mean they're independent? Suppose we have two events, $A$ and $B$. I want to find out if $A$ and $B^c$ are which one of mutually exclusive or independent. I have figured out that they are not mutually exclusive.. does that mean they are definitely independent? If so, how would I justify this?
 A: Suppose for your two events $A$ and $B$, $P(A)\ne 0$ and $P(B)\ne 0$.
Now suppose  $A$ and $B$ are mutually exclusive 
which means  $A\cap B =\varnothing$  then $P(A\cap B)=0 \tag{1}$
If $A$ and $B$ are independent then
$P(A\cap B)=P(A)P(B)\ne0 \tag{2}$ 
since neither $P(A)$ nor $P(B)$ are $0$.
You can see $(1)$ and $(2)$ are contradictory.
So you can see if $A$ and $B$ are mutually exclusive then, under the additional condition that $P(A)$ and $P(B)$ are both non-zero, they are in fact dependent i.e $P(A\cap B)\ne P(A)P(B)$.
One example is a disease and its vaccine.
Suppose you have a 100% effective vaccine such that you cannot contract the disease if you were vaccinated. Then being vaccinated and contracting the disease are mutually exclusive and they are dependent.
A: *

*Event that can't occur simultaneously are mutually exclusive. Eg. If you toss a coin you will get head or tail, so event of getting head and event of getting tail are mutually exclusive.

*If events are independent outcome of one does not affect other event.Eg. 2 different events of tossing two coins are independent as outcome of one does not affect other.


So mutual dependence is prerequisite for mutual exclusivity therefore mutually exclusive events can't be independent.
PS. These are layman definitions might not be mathematically correct.
A: No. You can have dependent events that are not mutually exclusive.
Consider the events:
$A$: The radio traffic report says that traffic is "heavy".
$B$: I am late for work
$B^c$: I am not late for work
Neither $B$ nor $B^c$ are independent of $A$ (since I am more likely to be late when the radio says traffic is heavy than otherwise), but neither is mutually exclusive of it since I would generally still be there in sufficient time, but will not always be there on time irrespective of $A$ happening.
[To be independent, two events with non-zero probability cannot be mutually exclusive, but the converse doesn't hold.]
A: Consider this example:
Rolling the dice only once:

*

*Event A: The result is 1 or 6.

*Event B: The result is 1 or 3.


*

*Does Event A and Event B mutually Exclusive? No, two events occur simultaneously if the result is 1.

*Does Event A and Event B independent? No, because if event A occurs, it increases the chance that event B occurs.

So does "not mutual exclusive" implies independence? NO!
