Confused About Mutual Exclusivity with More Than Two Events

Spud

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Hello,

Sorry if this is the wrong section of the forums, but I figured that questions about mutually exclusive events are relevant to probability.

My current understanding:
Two events are mutually exclusive if both events cannot occur at the same time. In other words, two events are mutually exclusive if the probability of both events occurring at the same time is 0.

I guess I'll use an example with fair six-sided dice to try and explain where my confusion lies.

Two Events: The event Roll 1 and the event Roll 3 or 4 are mutually exclusive (none of the six outcomes belong to both events).
Two Events: The event Roll 3 or 4 and the event Roll 4 or 5 are not mutually exclusive (the outcome of 4 belongs to both events).

My question:
Are the three events Roll 1, Roll 3 or 4 and Roll 4 or 5 considered to be mutually exclusive or not mutually exclusive?

On one hand, the three events seem to be not mutually exclusive because two of the three events can occur at the same time. But on the other hand, the three events seem to be mutually exclusive because the probability of all three events occurring at the same time is 0.

Can someone please advise me where the mistake in my thinking lies? Perhaps I'm not using a good definition of mutual exclusivity? Is there a standard definition that I should be using?

Thanks a lot!
 
My current understanding:
Two events are mutually exclusive if both events cannot occur at the same time. In other words, two events are mutually exclusive if the probability of both events occurring at the same time is 0.
I guess I'll use an example with fair six-sided dice to try and explain where my confusion lies.

Two Events: The event Roll 1 and the event Roll 3 or 4 are mutually exclusive (none of the six outcomes belong to both events).
Two Events: The event Roll 3 or 4 and the event Roll 4 or 5 are not mutually exclusive (the outcome of 4 belongs to both events).

My question:
Are the three events Roll 1, Roll 3 or 4 and Roll 4 or 5 considered to be mutually exclusive or not mutually exclusive?
On one hand, the three events seem to be not mutually exclusive because two of the three events can occur at the same time. But on the other hand, the three events seem to be mutually exclusive because the probability of all three events occurring at the same time is 0.
There does not seem to be universal agreement on the exact definition.
That said, here is a link that talks about several understandings.
 
There does not seem to be universal agreement on the exact definition.
That said, here is a link that talks about several understandings.

Hey pka, thanks for your response and link.

The problem that I've found is that almost every source that provides a definition is strictly speaking within the context of two events. So when I extend the scenario to more than two events, different definitions start to produce different results when classifying things as ME or not ME.

So, are you essentially saying that it entirely depends on the definition one is working with? That's fine with me, but I guess I was hoping for some kind of "official" definition used in academic mathematics?

Cheers!
 
Excellent question! I was taught that the definition of ME meant that if you had n events, E1, E2,...,En, then Ei intersect Ej is empty if i and j were different. This is called pairwise disjoint (if I remember correctly). BUT this is textbook/instructor dependent. There is no universal agreement with this. This happens with many mathematical definitions.
 
Excellent question! I was taught that the definition of ME meant that if you had n events, E1, E2,...,En, then Ei intersect Ej is empty if i and j were different. This is called pairwise disjoint (if I remember correctly). BUT this is textbook/instructor dependent. There is no universal agreement with this. This happens with many mathematical definitions.

Hey Jomo, thanks for your input!

Here are the two most common definitions that I've found:

Definition 1:
A set of events are mutually exclusive if the occurrence of one event in that set precludes the occurrence of all other events in that set.

With this definition, the three events in my example are not mutually exclusive.

Definition 2:
A set of events are mutually exclusive if the probability of all events in that set occurring at the same time is 0.

With this definition, the three events in my example are mutually exclusive.

The way you've been taught about ME is I think is more akin to definition 1. But yeah, when I get into discussions about mutually exclusive events and mutually inclusive events, everything is fine if we're just talking about two events, but as soon as a third event comes into play, it changes everything and disagreements start rapidly appearing.

I guess I'm wondering which definition is correct and why? Perhaps there is a certain context (probability, logic, statistics, etc) in which each definition is more suited to? I have no idea! I just wish there was a solid, clear definition!
 
The problem that I've found is that almost every source that provides a definition is strictly speaking within the context of two events. So when I extend the scenario to more than two events, different definitions start to produce different results when classifying things as ME or not ME.

So, are you essentially saying that it entirely depends on the definition one is working with? That's fine with me, but I guess I was hoping for some kind of "official" definition used in academic mathematics?
As the Wikipedia article says, "To say that more than two propositions are mutually exclusive, depending on context, means [either] that one cannot be true if the other one is true, or [alternatively, that] at least one of them cannot be true. The term pairwise mutually exclusive always means that two of them cannot be true simultaneously." So this is context-dependent.

I suspect that the reason most definitions (including the main discussion there) refer specifically to two events is that the term is usually used only in such contexts. When terms are not used often in a certain context, there typically is no consensus, and it is standard practice to state one's own definition as applied to that context (in this case, more than two events).

So if (as suggested by your clause, "when I extend the scenario") you yourself want to write a paper about more than two events using this term, you are free to define how you are using it. More than that, you are required to state how you are using it, both for the sake of your readers, and in order to ensure that your own reasoning is consistent. Or, as Wikipedia suggests, you can insert the word "pairwise" to remove uncertainty. Or you can just start off by saying, "When I use the term "mutually exclusive", I will always mean pairwise mutually exclusive."

On the other hand, if you are asking because you have seen the term applied to three or more events, it would be very helpful if we could see that source -- partly to determine whether it states or implies a definition, and partly just to give us a specific example of such usage to discuss.
 
I guess I'm wondering which definition is correct and why? Perhaps there is a certain context (probability, logic, statistics, etc) in which each definition is more suited to? I have no idea! I just wish there was a solid, clear definition!
I guess I'm wondering which definition is correct and why? No need to guess as they are both correct (just not at the same time).
There is no need for arguments between two people as you must first decide on a definition!
 
As the Wikipedia article says, "To say that more than two propositions are mutually exclusive, depending on context, means [either] that one cannot be true if the other one is true, or [alternatively, that] at least one of them cannot be true. The term pairwise mutually exclusive always means that two of them cannot be true simultaneously." So this is context-dependent.

I suspect that the reason most definitions (including the main discussion there) refer specifically to two events is that the term is usually used only in such contexts. When terms are not used often in a certain context, there typically is no consensus, and it is standard practice to state one's own definition as applied to that context (in this case, more than two events).

So if (as suggested by your clause, "when I extend the scenario") you yourself want to write a paper about more than two events using this term, you are free to define how you are using it. More than that, you are required to state how you are using it, both for the sake of your readers, and in order to ensure that your own reasoning is consistent. Or, as Wikipedia suggests, you can insert the word "pairwise" to remove uncertainty. Or you can just start off by saying, "When I use the term "mutually exclusive", I will always mean pairwise mutually exclusive."

On the other hand, if you are asking because you have seen the term applied to three or more events, it would be very helpful if we could see that source -- partly to determine whether it states or implies a definition, and partly just to give us a specific example of such usage to discuss.

Hey, thanks for your input; it's appreciated.

So, after asking around on a bunch of forums, the following seems to be agreed upon the most:

Event 1: Roll 1
Event 2: Roll 3 or 4
Event 3: Roll 4 or 5

Event 1 and Event 2 are mutually exclusive.
Event 1 and Event 3 are mutually exclusive.
Event 2 and Event 3 are not mutually exclusive.

Even though the first two pairs of events are mutually exclusive, the set of all three events are not mutually exclusive simply due to the fact that the third pair of events are not mutually exclusive. Basically, if any two events within a set are not mutually exclusive, then the entire set of events is not mutually exclusive.

This means that if we changed Event 3 to Roll 5, then the set of three events would be mutually exclusive.

What are your thoughts on this?
 
I guess I'm wondering which definition is correct and why? No need to guess as they are both correct (just not at the same time).
There is no need for arguments between two people as you must first decide on a definition!

I completely agree. I just posted a response to Dr.Peterson's comment that contains my updated understanding of this situation, based on what a bunch of others have said on various forums.

The consensus seems to be that the three events are not mutually exclusive.

What are your thoughts?
 
My own general sense is that mutually exclusive means pairwise disjoint, which is your Definition 1. In fact, to my mind, "mutually" essentially means "pairwise": each one excludes each of the others. So I agree with what you said about your three events.

Your Definition 2 is wrong in the first place because it is not about probability, but about possibility. In some contexts, you can have probability zero although the events have a nonempty intersection. What you mean there is that the intersection of the sets is empty. And that is not what I mean by mutually exclusive.

I'm not sure I really know what the Wikipedia quote is saying; I'm not positive that it really says there are different definitions. But I just noticed that below that quote, under Probability, they do make a definite statement: "In probability theory, events E1, E2, ..., En are said to be mutually exclusive if the occurrence of any one of them implies the non-occurrence of the remaining n − 1 events."

What I am curious about still is, where did you find the other definitions? You have mostly just referred to people arguing, apparently based on their own understandings. If you actually found reference sites stating Definition 2, I want to see their context.
 
My own general sense is that mutually exclusive means pairwise disjoint, which is your Definition 1. In fact, to my mind, "mutually" essentially means "pairwise": each one excludes each of the others. So I agree with what you said about your three events.

At this point in time, we are in complete agreement. Although, just recently someone posted on another forum stating that he disagrees because he uses the following definition of mutual exclusivity:

Alternative definition:
A set of events are mutually exclusive if at least one pair of events within the set are mutually exclusive.

Whereas we are currently using this definition (or something similar):
A set of events are mutually exclusive if all pairs of events within the set are mutually exclusive.

These two definitions can produce different results, but I'm currently more inclined to follow the latter definition.

Your Definition 2 is wrong in the first place because it is not about probability, but about possibility. In some contexts, you can have probability zero although the events have a nonempty intersection. What you mean there is that the intersection of the sets is empty. And that is not what I mean by mutually exclusive.

Good point. But what if we changed the second definition to:

Modified Definition 2:
A set of events are mutually exclusive if it's impossible for all events in the set to occur at the same time.

With this modified definition of Definition 2, my example of three events would be considered mutually exclusive.

I'm not sure I really know what the Wikipedia quote is saying; I'm not positive that it really says there are different definitions. But I just noticed that below that quote, under Probability, they do make a definite statement: "In probability theory, events E1, E2, ..., En are said to be mutually exclusive if the occurrence of any one of them implies the non-occurrence of the remaining n − 1 events."

It would seem that the main difference -- to the extent that there even is a difference -- is between the context of probability and the context of logic.

What I am curious about still is, where did you find the other definitions? You have mostly just referred to people arguing, apparently based on their own understandings. If you actually found reference sites stating Definition 2, I want to see their context.

Sorry if I didn't explain this properly, but yes, what I was referring to was people disagreeing amongst themselves due to different understandings of the term and due to almost never having to actually apply the term to more than two events.

Having said that, however, I have found some sources that seem to be advocating for Definition 2 (or something similar):

1. https://courses.lumenlearning.com/a...er/independent-and-mutually-exclusive-events/

A and B are mutually exclusive events if they cannot occur at the same time.
This means that A and B do not share any outcomes and P(A AND B) = 0.

2. https://www.mathsisfun.com/data/probability-events-mutually-exclusive.html

When two events (call them "A" and "B") are Mutually Exclusive it is impossible for them to happen together: P(A and B) = 0
"The probability of A and B together equals 0 (impossible)"

3. https://www.statisticshowto.datasciencecentral.com/mutually-exclusive-event/

Mutually exclusive events are things that can’t happen at the same time.

4. https://magoosh.com/statistics/mutually-exclusive-events-definition-and-examples/

If two events are mutually exclusive, it means that they cannot occur at the same time.

5. https://www.toppr.com/guides/quantitative-aptitude/probability/mutually-exclusive-events/

Two events are said to be mutually exclusive events when both cannot occur at the same time.

6. https://www.onlinemathlearning.com/mutually-exclusive-events.html

Two events are said to be mutually exclusive if they cannot happen at the same time.

Most -- if not all -- of these sources are speaking within the context of only two events. So it doesn't really matter which definition is used when comparing two events because the result is always the same. But if we use these definitions for three or more events, my example of three events would be considered mutually exclusive. If I am understanding this incorrectly, please let me know.

I also found some sources that seem to be advocating for Definition 1 (or something similar):

1. https://www.intmath.com/counting-probability/9-mutually-exclusive-events.php

Two or more events are said to be mutually exclusive if the occurrence of any one of them means the others will not occur (That is, we cannot have 2 or more such events occurring at the same time).

2. http://mathworld.wolfram.com/MutuallyExclusiveEvents.html

n
events are said to be mutually exclusive if the occurrence of any one of them precludes any of the others.

3. https://www.investopedia.com/terms/m/mutuallyexclusive.asp

Mutually exclusive is a statistical term describing two or more events that cannot coincide. It is commonly used to describe a situation where the occurrence of one outcome supersedes the other.

All in all, I'd say that my lack of understanding stems for a large variety of sources using different definitions that -- due to only providing examples of two events -- result in inconsistencies and my confusion when extending the scenarios to three or more events.
 
The comment is not an "I told you so". However, this is a can of worms.
Once I was chairing a state wide session on teaching probability & stat. I thought we would come to blows over what it means for a collection of events to be independent. Though that is not "exclusive" it is nonetheless similar.
 
Most -- if not all -- of these sources are speaking within the context of only two events. So it doesn't really matter which definition is used when comparing two events because the result is always the same. But if we use these definitions for three or more events, my example of three events would be considered mutually exclusive. If I am understanding this incorrectly, please let me know.
...
All in all, I'd say that my lack of understanding stems for a large variety of sources using different definitions that -- due to only providing examples of two events -- result in inconsistencies and my confusion when extending the scenarios to three or more events.
It appears that you have no references (except maybe your quote #3) to the term applied definitely to more than two events, that state anything other than the pairwise (#1) definition. You are only extrapolating, assuming that the wording "can't occur at the same time" would still be used in the case of more than two. It's quite common for definitions or rules to have to be reworded when extending from two to more than two; for example, the rule for signs of products of two numbers can be stated as "positive if the signs are the same", but for more than one, this has to be changed to "positive if there are an even number of negatives". Similarly, the relationship between GCF and LCM is simple for two numbers, but much more complicated for more. Extrapolation is not a good way to confirm definitions.

But the main point is that the term is almost always used of only two events (or with the qualifier "pairwise"), so the best course is to (a) avoid using the term for larger sets of events, and (b) state your definition when you need to. Arguments aren't needed; clarity is. Claims about definitions need to be justified by evidence of common usage, not by saying "this definition makes more sense"; if the latter is true, fine, but just state that as your definition, and see if the rest of the world follows.
 
It appears that you have no references (except maybe your quote #3) to the term applied definitely to more than two events, that state anything other than the pairwise (#1) definition. You are only extrapolating, assuming that the wording "can't occur at the same time" would still be used in the case of more than two. It's quite common for definitions or rules to have to be reworded when extending from two to more than two; for example, the rule for signs of products of two numbers can be stated as "positive if the signs are the same", but for more than one, this has to be changed to "positive if there are an even number of negatives". Similarly, the relationship between GCF and LCM is simple for two numbers, but much more complicated for more. Extrapolation is not a good way to confirm definitions.

But the main point is that the term is almost always used of only two events (or with the qualifier "pairwise"), so the best course is to (a) avoid using the term for larger sets of events, and (b) state your definition when you need to. Arguments aren't needed; clarity is. Claims about definitions need to be justified by evidence of common usage, not by saying "this definition makes more sense"; if the latter is true, fine, but just state that as your definition, and see if the rest of the world follows.

This is why I am here; I want to find answers and I want to learn. I am not assuming anything; I'm asking for explanations and clarifications from people with more experience and knowledge than me. I just want to clearly understand. If I was assuming the answer, then I wouldn't be here.

I suppose I'm just confused as to why the seemingly large majority of sources are giving definitions that exclusively apply to two events and strictly don't apply to more than two events (without even making this clear, in most cases).

Before I came to this website, I simply couldn't nail down a coherent definition; so I decided that I needed help. I was confused and frustrated that there doesn't seem to be a universal definition that always applies, irrespective of the set containing two, three or 500 events.

I understand your points and I agree with you.

Side note:
If a set of events are not mutually exclusive, does that mean the set of events are mutually inclusive?
 
The comment is not an "I told you so". However, this is a can of worms.
Once I was chairing a state wide session on teaching probability & stat. I thought we would come to blows over what it means for a collection of events to be independent. Though that is not "exclusive" it is nonetheless similar.

Would you be willing to explain independent events to me? I'd love to learn about it!
 
Spud, as I said before, your question is an excellent one. I think that just because you were concerned enough about a definition and asked for an answer on this site. Good for you!

However I think that you are spending too much time on This. After all, it is not a concept you are trying to learn. People on this site would help you until you finally understood a concept. But this is a definition! Maybe in writing a research paper a mathematician would benefit by using one definition of ME over another. Other times the mathematician would use the other definition. I am just trying to suggest that you not get caught up over the fact that there is more than oine definition for ME. It is so wonderful that you know this and now know to ask when you are talking to someone about ME which definition they are using and then proceed with the discussion.

On this website some students in advanced algebra ask for help with composition of function like f o g o h. Do you know some authors/mathematicians don't agree on the order of composition. So when I see a question like this my 1st question to the poster is to state the order they are using or give me a worked out example so I can figure out the order. I just respect the instructor or author and do it that way. I don't get hung up on the fact that the definition is not universal.

Sure I wish things were more universal but they just are not. Please think about what I am saying.
 
Would you be willing to explain independent events to me? I'd love to learn about it!
Two events are said to be independents if the outcome of one event does not affect the outcome of the 2nd event.

Tossing a coin is independent because the prior tosses does not change the change the fact that the probability of tossing a head with a fair coin is 1/2. Even if you tossed a fair coin 1000 times and got 998 head, the chance of getting a head on the next toss is still 1/2. Same story with tossing a coin and rolling a die.
 
Would you be willing to explain independent events to me? I'd love to learn about it!
I will show you how mathematicians fight over words. Have you ever been told that a prime number is a positive integer which is divisible only by itself and \(\displaystyle \bf1\). Well by that definition \(\displaystyle 1\) is a prime. \(\displaystyle 1\) is divisible by itself & \(\displaystyle 1\) hence it is prime. But of course \(\displaystyle 1\) is not prime. So how do we correct the definition: a prime number is a positive integer which has exactly two divisors. How does that exclude \(\displaystyle 1\) from being prime by that definition?
Those of us who are second-generation dependents of R L MOORE also see HERE
Moore is an interesting study in history of mathematics. In the late 1890's he was a high school geometry teacher in rural East Texas. Veblan or EH Moore(no relation), one of them got word that RL had shown that Hilbert's axioms for geometry were redundant. So they brought him to U of Chicago. All he had to do was the basic required courses for the PhD. R L Moore became the father of American topology and as Devlin the PRM's mathguy says RLMoore was the most influential person on twentieth century mathematics for higher education.
 
However I think that you are spending too much time on This. After all, it is not a concept you are trying to learn. People on this site would help you until you finally understood a concept. But this is a definition!

Hey Jomo,

Thanks for your input.

I agree with almost everything you've said here, except for one thing (quoted above) which I suspect we actually do agree on, but perhaps I just misunderstood you or perhaps you misspoke.

Essentially, what I am trying to do is to understand a concept. But every concept is hidden behind a label and every label comes with a definition to explain that concept.

Before I came to the website, I was under the impression that there was one definition of mutual exclusivity to explain one concept of mutual exclusivity. However, as we are all aware, I have since learnt that there are at least two (and perhaps even more) different definitions of mutual exclusivity to explain two slightly different concepts of mutualy exclusivity.

The biggest takeaway for me from talking about this with you guys is that when discussing mutual exclusivity (or any topic) with people, we need to nail down the actual concept we are talking about, coupled with an agreed-upon definition of that concept, at the beginning of the discussion.

Being clear about the difference between two events and more than two events is very important (which is the thing that has frustrated me the most, as a lot of the sources that provide definitions don't actually make this point clear). Going through all this step-by-step at the beginning would almost certainly avoid confusion later on in the discussion.

Sure I wish things were more universal but they just are not. Please think about what I am saying.

Yep, I totally understand and I think we are in agreement.

While you're up and about, can I ask you to explain mutual inclusivity?

I was under the impression that mutual exclusivity and mutual inclusivity were logical opposites. As in, if a set of events are not ME, then they are MI; and vice versa. But now I'm almost certain that this is not the case at all.

Which poses the question:
If a set of events is not ME and not MI, then what are they? Is there a term for a set of events that are neither ME nor MI?

Thanks!
 
Two events are said to be independents if the outcome of one event does not affect the outcome of the 2nd event.

Tossing a coin is independent because the prior tosses does not change the change the fact that the probability of tossing a head with a fair coin is 1/2. Even if you tossed a fair coin 1000 times and got 998 head, the chance of getting a head on the next toss is still 1/2. Same story with tossing a coin and rolling a die.

Alrighty, makes sense; thanks.

But just to make sure I understood this correctly...

Picking one random card (with replacement) from a standard deck of shuffled playing cards 20 times are independent events.
But picking one random card (without replacement) from a standard deck of shuffled playing cards 20 times are dependent events.

Right? :unsure:
 
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