Given Events A and B If independent events means that the probability of event A occurring does not affect the probability of B occurring and mutually exclusive events means that A and B does not intersect - they cannot occur at the same time. I understand that mutually exclusive events mean that they are dependent.
Given that we know 2 events are independent, what does that tell us about them being disjoint or not disjoint?
Independence intuitively tells me 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.
