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Google says, "mutually exclusive is a statistical term describing two or more events that cannot happen simultaneously."

But when we say that $n$ events are mutually exclusive, we mean that nothing is in common between them, correct?

So, when we say that $n$ claims are mutually exclusive, are we actually saying that exactly 1 is true, or that at most 1 is true?

ryang
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Bob Marley
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    It means that if one of the claims is true, then all of the others must be false. It is perfectly possible for all the claims to be false. – lulu Jun 02 '24 at 20:41
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    at most one is true. – Shraiysh Jun 02 '24 at 20:45
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    But when we say "n mutually exclusive events", we mean to say nothing is in common between them, correct? – Bob Marley Jun 02 '24 at 20:45
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    Not following. With mutually exclusive events, we again have that, if one of them occurs, none of the others occur. So, again, at most one of the events might occur. – lulu Jun 02 '24 at 20:47
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    Sometimes one speaks of events which are both "mutually exclusive" and "exhaustive". That means that exactly one of the events must occur. If you select a natural number, for example, it must be even or odd, it can't be both. – lulu Jun 02 '24 at 20:48
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    Sorry I mean, as per google definition, "Mutually exclusive is a statistical term describing two or more events that cannot happen simultaneously"... so if 2 or more things can't occur simultaneously, then that means $\geq 2$ events can't be "true" (or occuring) at same time, hence it's "at most 1 event" is occuring/true? – Bob Marley Jun 02 '24 at 20:49
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    @lulu Oh ok I see. So "mutually exclusive" means "$\leq1$ event occurs/is true" and "exhaustive" means "$\geq1$ event occurs/is true"? – Bob Marley Jun 02 '24 at 20:51
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    That's right. $\quad$ – lulu Jun 02 '24 at 20:57
  • Let us continue this discussion in chat. – Bob Marley Jun 02 '24 at 21:00

2 Answers2

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With reference to three or more events, there are actually two conflicting definitions of mutually exclusive. Consider the mutually exclusive events $A, B$ and $C.$

Google says, "mutually exclusive is a statistical term describing two or more events that cannot happen simultaneously."

By this definition (collectionwise mutual exclusivity), no outcome belongs to all of the three events, that is, $$A\cap B\cap C=\emptyset.\tag1$$

But when we say that $n$ events are mutually exclusive, we mean that nothing is in common between them, correct?

By this definition (pairwise mutual exclusivity), no pair of events can simultaneously happen, that is, $$A\cap B=B\cap C=A\cap C=\emptyset.\tag2$$

So, when we say that $n$ claims are mutually exclusive, are we actually saying that exactly 1 is true, or that at most 1 is true?

$\text“n$ claims are mutually exclusive”, by the first definition:

  • at least one claim is false.

$\text“n$ claims are mutually exclusive”, by the second definition ✔️:

  • at most one claim is true.

$\text“n$ claims are collectively exhaustive”:

  • at least one claim is true.

$\text“n$ claims are mutually exclusive and collectively exhaustive”, by the second definition ✔️:

  • exactly one claim is true.
ryang
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When we say that n claims are mutually exclusive, we say that only 1 of the n claims are going to be true unless none are true. Imagine mutually exclusive events being a roulette wheel in a casino, where you only have 1 ball and each of the slots that the ball can go into are mutually exclusive events, the ball can only go into 1 of the slots.

Ibrokhim Bakirov
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