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

disjoint: P(A intersection B) = 0 <=> P(A union B) = P(A) + P(B)

independent: P(A 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...

  • $\begingroup$ Welcome to MSE. Please use MathJax to format your posts; you'll find you get a much better response if your questions are easy to read. $\endgroup$
    – saulspatz
    Sep 22 '20 at 22:15
  • 1
    $\begingroup$ The statements “disjoint events with positive probability are not independent” and “independent events with positive probability are not disjoint” are equivalent statements. You have proven the first statement, therefore the second one is true. $\endgroup$ Sep 22 '20 at 22:54

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.

  • $\begingroup$ Good point! It’s easy to get lost in the formalism without thinking about what’s actually going on $\endgroup$ Sep 23 '20 at 10:32

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.

  • $\begingroup$ I don't know if I misunderstood, but my question was for the other way around... Can two independent events with positive probabilities be disjoint? $\endgroup$
    – asprog
    Sep 22 '20 at 22:34
  • $\begingroup$ @asprog Fixed answer! $\endgroup$ Sep 22 '20 at 22:57
  • $\begingroup$ thank you so much for the help!! $\endgroup$
    – asprog
    Sep 22 '20 at 23:29

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