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It is not clear to me what is/should be the standard definition of "mutually exclusive" in probability, as there seem to be two definitions in the literature.

  1. In the top response in this thread, mephistolotl wrote

Two events are mutually exclusive if the probability of them both occurring is zero, that is if $\text{Pr}(A\cap B)=0$.

The user also said that this is the definition in some but not all texts.

  1. On the other hand, the Wikipedia article on mutual exclusivity says that

Formally said, the intersection of each two of them is empty (the null event): $A\cap B= \varnothing$.

Does one definition dominate the other in mathematics, and if so, which one? If not, what are the merits and disadvantages of each? Of course, we could just give a different name to each of the two, but given the prevalence of the term "mutually exclusive," I am interesting in knowing the best meaning to assign to it.

If it helps, I am mainly interested in discrete probability at the moment, but it would be nice if the definition extended to general probability. In discrete probability, if we know that none of the elements have zero probability, then the two definitions are equivalent.

Favst
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    $A\cap B=\emptyset \implies Pr(A\cap B)=0$ but not the other way around. It is possible to have events occur almost never (see almost everywhere on wikipedia). I generally see $A\cap B=\emptyset$ specifically to be the definition of mutual exclusivity – JMoravitz Apr 28 '20 at 18:35

1 Answers1

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I think it's better to be consistent with natural language and define mutually exclusive events to be events $A$ and $B$ such that $A\cap B=\emptyset$. If you look at Merriam-Webster, the term "mutually exclusive" is defined as being related such that each excludes or precludes the other. This is not technically true for events that merely intersect in a probability $0$ event. If two events intersect in a nonempty event with probability $0$, an event contained in their intersection can occur in a non-discrete probability space, so neither event would necessarily exclude or preclude the other.

For example, the events $[0,1]$ and $[1,2]$ intersect in $\{1\}$, and, say in the uniform distribution on $[0,2]$, this intersection has probability $0$. However, drawing a sample from the uniform distribution on $[0,2]$ could certainly result in a value of $1$, even though the event has probability $0$.

The only virtue I see of using the alternative definition of mutually exclusive is that we have a more general situation over which the probability of the union of two events is the sum of the probabilities of the events. I don't see this as worth throwing out the consistency of the term with natural language, though.

Matt Samuel
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  • I don't quite see how the example that you provided argues that one of the definitions is preferable to the other. Maybe it's because I did not fully understand what you meant by "natural language" and "logical consistency of the term". – Favst Apr 28 '20 at 19:05
  • @Favst I updated my answer. Let me know if that's clearer. – Matt Samuel Apr 28 '20 at 19:10
  • Makes sense now! – Favst Apr 28 '20 at 19:22