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Larson (1982) defining the probability axioms talks about "mutually exclusive" events, while Poirier (1995) about "$A_1, A_2, \ldots$ as a sequence of pairwise mutually exclusive events events in the sigma-algebra $\tilde A$."

I suppose that the two notions are equivalent (they both refer two disjoint sets), right? Does this make adding the word "pairwise" superfluous on behalf of Poirier?

Is there any other context out of probability that makes this distinction (using the word pair-wise) meaningful? According to wikipedia, in Logic, "pairwise mutually exclusive" means that both propositions cannot be true simultaneously, in contrast to just mutually exclusivity that means that if one is true, then the other cannot be true.

Carl Mummert
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Konstantinos
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5 Answers5

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What about vicious cycles? For example, suppose you have three binary random variables $A$, $B$, and $C$ that each take values $0$ or $1$. In this example, $A$ and $B$ are mutually exclusive if whenever $A = 0$, $B = 1$, and similarly for $C$.

$A$, $B$, and $C$ cannot all be pairwise mutually exclusive.

However, they can be "globally" mutually exclusive in the sense that, for example, if $A = 0$, then $B = 1$ and $C = 1$, and so on for the other variables.

I think that this counterexample proves that pairwise mutual exclusivity is not equivalent to mutual exclusivity.

gideonite
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Pick a three-letter word from an English dictionary. Let event A be, the word contains an "a"; let event E be, the word contains an "e"; let event I be, the word contains an "i".

A and E are not mutually exclusive ("are" contains both "a" and "e").

A and I are not mutually exclusive ("air" ....).

E and I are not mutually exclusive ("ire" ....).

So the three are not pairwise mutually exclusive, in fact, no pair is mutually exclusive. But the three are mutually exclusive, since no three-letter word contains all three letters.

Gerry Myerson
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  • I (like this answer and wikipedia) believe that the mutual exclusivity of multiple events requires all pairwise joint probabilities to be zero; in fact this is a sufficient condition, because it trivially implies that all joint probabilities (i.e., that of each trio of events, and so on) are zero (this is unlike for independence, where pairwise independence doesn't imply mutual independence). However, both your example and gideonite's are using a different definition of ME. – ryang Feb 04 '23 at 19:06
  • Picking a number uniformly at random from the interval $[0,1]$, the events "the number is rational" and "the number exceeds one-half" are mutually exclusive by the joint-probabilities-zero definition, since "the number is rational" already has probability zero. Indeed, by that definition, an event will be mutually exclusive with itself, provided it is a probability zero event. Strikes me as a peculiar definition. (And I think Wikipedia agrees.) – Gerry Myerson Feb 04 '23 at 21:36
  • Yes, the peculiarity that you are observing is due to defining ME using joint probability instead of disjointedness (luckily, experiments on finite sample spaces aren't affected). $\quad$ However, I was making a separate point, that your preferred definition of ME (and since the sample space for your AEI example is finite, I will just use the more straightforward disjointedness defn over the the joint-probability defn) merely asks that $A\cap E\cap I=\emptyset,$ whereas my definition (like wikipedia's) requires that each pair of $A,E,I$ is disjoint (so, this defn says that $AEI$ aren't ME). – ryang Feb 05 '23 at 13:29
  • Hehe, chatGPT is self-contradictorily claiming both our definitions (note that the word "meaning" that follows is asserting a false equivalence): "Three events are mutually exclusive if they cannot occur simultaneously [your definition], meaning only one of the events can occur at a time [my definition]. In other words, if the occurrence of one event precludes the occurrence of any of the other events, then the events are considered to be mutually exclusive [my definition]." – ryang Feb 05 '23 at 13:44
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    @ryang, thanks, I misunderstood your point, now I see what you are getting at. – Gerry Myerson Feb 05 '23 at 20:27
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Larson (1982) talks about "mutually exclusive" events, while Poirier (1995) about "pairwise mutually exclusive events"

I suppose that the two notions are equivalent (they both refer two disjoint sets), right? Does this make adding the word "pairwise" superfluous on behalf of Poirier?

For simplicity, let's require the sample space to be finite and contain only possible outcomes, so that an empty event is precisely one that has zero probability (i.e., $P(S)=0\iff S=\emptyset$). With this, there appears to be two inequivalent definitions of mutual exclusivity (they converge only when discussing exactly two events):

  1. Gerry's and gideonite's answers on this page define multiple events as mutually exclusive iff no outcome belongs to all of them (e.g., for three events: $A\cap B\cap C=\emptyset$). In other words, a collection of mutually exclusive events is precisely a collection of events that cannot simultaneously occur.

    [i.e., collectionwise mutual exclusivity]

  2. This answer, this answer, Wikipedia and mine define multiple events as mutually exclusive iff they are pairwise disjoint. In other words, a collection of mutually exclusive events is precisely a collection of events that can occur only one at a time.

    [i.e., pairwise mutual exclusivity]

Definition 2 is stricter than Definition 1: observe that only the latter allows mutually exclusive events $A,B,C$ to be such that $A\cap B\ne\emptyset$ and such that $P(A)+P(B)+P(C)>1.$

Definition 2—but not Definition 1—is consistent with the definition of mutual independence, and the usual meaning of phrases like ‘mutual respect’, where mutual implies pairwise.

(However, note that while pairwise independence does not imply mutual independence, pairwise mutual exclusivity does imply both the above definitions of mutual exclusivity.)

P.S. Please refer to this answer for a parallel discussion of set disjointedness: Does ‘disjoint’ mean pairwise or collectionwise?. And in the answer ‘each’, ‘every’, ‘any’, ‘all’, I point out how the poor choice of the phrase ‘if any’ in Wikipedia's definition of the term ‘disjoint’ does not help disambiguate its definition!

ryang
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They don't mean the same thing, but it is a short jump to show they are equivalent. A set of events, $A_1,...,A_n$ are mutually exclusive if the occurrence of one of them implies that the other $n-1$ events can't happen. It's immediate that the events are pairwise mutually exclusive. The other direction is immediate as well.

They aren't a priori the same, but it's immediate that they're equivalent.

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Sets $A_1, A_2, ...$ are mutually disjoint if

$$\bigcap_{i=1}^{\infty} A_i = \emptyset $$

Sets $A_1, A_2, ...$ are pairwise disjoint if

$$A_i \cap A_j = \emptyset \ \forall i \ne j$$

Apparently, most texts use 'disjoint' to refer to 'pairwise disjoint'. Whenever a text uses 'pairwise disjoint', we can assume 'disjoint' refers to 'mutually disjoint'

In our case, 'mutually exclusive' by Larson means the same thing as 'pairwise mutually exclusive'. It depends on the text I guess.

See my questions:

Do Kolmogorov's axioms really need only disjointness rather than pairwise disjointness?

http://meta.math.stackexchange.com/questions/21560/should-these-be-simply-disjoint-instead-of-pairwise-disjoint

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