- Let say we made experiment that gave us N results.
- Let say m results are favorable for event A.
- Let say n results are favorable for event B.
Then we could picture it like that for independent, depended, mutually exclusive, and mutually events: events
According to the picture:
- independent and mutually exclusive events are the same thing
- dependent and mutually events are the same thing
Questions:
What the difference between:
- independent and mutually exclusive events
- depended and mutually events
Is it possible that event will be independent and mutually exclusive at the same time? If yes how we can use it?
Is it possible that event will be dependent and mutually at the same time? If yes how we can use it?
Two events $A,B$ are mutually exclusive iff: $$A\cap B=\varnothing\tag1$$
In words: it cannot happen that both events occur. At most one of them will occur, so the occurrence of $B$ excludes the occurrence of $A$ and vice versa. A consequence of this concerning probability is that: $$P(A\cap B)=P(\varnothing)=0$$
Two events $A,B$ are independent iff: $$P(A\cap B)=P(A)\times P(B)\tag2$$
A consequence of this is: $$P(A\mid B)=P(A)=P(A\mid B^{\complement})$$
In words: the probability of the occurrence of event $A$ is not depending on the occurrence (or non-occurence) of event $B$.
There is no mathematical concept "mutually events".
Two events $A$ and $B$ are mutually exclusive and independent at the same time iff :$$A\cap B=\varnothing\text{ and }0\in\{P(A),P(B)\}$$
In words: $A$ and $B$ have no outcome in common and at least one of them has probability $0$ to occur.