Can two independent events be disjoint?

Question

If events $A$ and $B$ both have positive probabilities, and if they are disjoint, they surely cannot be independent since:

$$\text{disjoint:}\quad P(A \text{ intersection } B) = 0 \iff P(A \text{ union }B) = P(A) + P(B),$$ $$\text{independent:}\quad\quad\quad\quad P(A \text{ intersection } B) = P(A)P(B).$$

so if P(A intersection B) is 0, then $P(A)P(B)$ should be 0 too, but since they're both above 0, then this is false.

However I am not sure if that is the case the other way around, I cannot put my head around the question if two independent events can be disjoint. Can anyone help? Thanks in advance.

Answer 1

Two independent events are disjoint only if at least one of them almost never happens.

More precisely: let $A, B$ be two independent events in the sample space $\Omega$ which are disjoint. Then $0 = P(A \cap B) = P(A) * P(B)$, so at least one of $A, B$ must have probability zero.

Answer 2

I agree with the other comments and answers. However I would have attacked the question solely by intuition: "if two events are independent, can they also be disjoint?

It is true that this problem can be attacked with math: assuming that events A and B each have a non-zero probability of occurring, they will be regarded as independent $\iff p(A) = p(A|B).$ Since A,B are disjoint, $p(A|B) = 0.$ Since it is assumed that $p(A) > 0, ~p(A) \neq p(A|B).$ Therefore, the two events can't be independent.

However, this problem can also be attacked by considering
event $C = $ the complement of event $B$
and showing, purely by intuition, that events $A$ and $C$ can not be independent.

Consider disjoint events A,B placed in a Venn diagram that represents the universe U.

Informally, $p(A)$ may be regarded as the proportion of the area assigned to event $A$ versus the area of the entire universe $U$ in the Venn diagram.

Since the event $C$ completely encompasses the event $A$, $p(A|C)$ may be similarly regarded as the proportion of the area assigned to event $A$ versus the area assigned to event $C$, rather than versus the area assigned to $U$.

Since $p(B)$ is assumed to be non-zero, the area assigned to event $C$ must be less than the area assigned to $U$. Therefore, the two proportions referred to in the above two paragraphs must be different.

Continuing this informal train of thought, suppose you have any two events $A$ and $B$, with $C$ = the complement of $B.$

Suppose further that $p(A) \neq 0, p(B) \neq 0, p(C) \neq 0.$

Further suppose that you have somehow concluded that events $A$ and $C$ are not independent. That means that the chance of $A$ occurring has been affected (i.e. altered) by whether it is to be assumed that event $C$ has also occurred.

It seems to me that if the chance of $A$ occurring has been affected by whether event $C$ has also occurred, then it is implied that the chance of $A$ occurring has also been affected by whether event $B$ has occurred.

In other words, when it is assumed that that $p(A) \neq 0, p(B) \neq 0,$ and $p(C) \neq 0,$ then regardless of any considerations of disjointness,
events $A$ and $B$ are independent $\iff$ events $A$ and $C$ are independent.