# The relationship between Mutually Exclusive and Independent and Disjointedness

Given events A and B, independence means that the probability of event A occurring does not affect the probability of B occurring, while mutually exclusivity means that A and B do not intersect (cannot occur at the same time).

I understand that mutually exclusive events are dependent.

Given that we know 2 events are independent, what does that tell us about them being disjoint or not disjoint? Intuitively, I feel that they are not disjoint.

If two events are independent, occurence of one does not have any affect on the probability of occurrence of the other. So, if $$A$$ and $$B$$ are independent $$P(A)=P(A|B)$$. Moreover, independent events can occur simultaneously; $$P(A\cap B)=P(A)P(B)$$ if A and B are independent. However, if two events are mutually exclusive, they cannot occur simultaneously so $$P(A\cap B)=0$$ if $$A$$ and $$B$$ are mutually exclusive. Consider the following example: flip a fair coin twice. Let F be the event that the first flip is a head, and S be the event that the second flip is a tail. Then, $$P(F)=P({H,H})+P({H,T})=1/2$$ and $$P(S)=P({H,T})+P({T,T})=1/2$$. Then, $$P(F\cap S)=P({H,T})=1/4=P(F)P(S)$$. Thus, $$F$$ and $$S$$ are independent and they have a nonempty intersection.

I understand that mutually exclusive events are dependent. Given that we know 2 events are independent, what does that tell us about them being disjoint or not disjoint? Intuitively, I feel that they are not disjoint.

For events with nonzero probability, independence $$\Big(\text{for two events: }P(A\cap B)=P(A)P(B)\Big)$$ implies not mutually exclusive $$\Big(\text{for two events: }P(A\cap B)\ne0\Big),$$ which in turn implies not disjoint.

while mutually exclusivity means that A and B do not intersect (cannot occur at the same time).

Not necessarily: here is a pair of non-disjoint, mutually exclusive events.

Given events A and B, independence means that the probability of event A occurring does not affect the probability of B occurring

But how does a probability—which is a value—affect another probability?