I often see that events are described as either mutually exclusive and independent events. My first question is, do mutually exclusive events simply refer to dependent events?
To add to the confusion, I also learned that mutually exclusive events can be independent event in a special case when probability is zero i.e. $P(A) = 0$ or $P(B) = 0$ .
But an event should either be independent or dependent and not both?
By definition, two events $A$ and $B$ are mutually exclusive whenever $A\cap B=\varnothing$. In that case, one has $\mathbb P(A\cap B)=0$. Some people also say that $A$ and $B$ are mutually exclusive whenever $\mathbb P(A\cap B)=0$. (The second condition is weaker, but it doesn't make a difference in practice.)
Now, two events $A$ and $B$ are called independent, if $\mathbb P(A\cap B)=\mathbb P(A)\cdot\mathbb P(B)$.
Can you now see in which cases you have that $A$ and $B$ are mutually exclusive and independent?
