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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.

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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.

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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?

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