Confused About Mutual Exclusivity with More Than Two Events

[Spud]

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!

 

[Ackbach]

According to the wiki article:

...events $E_1, E_2, \dots, E_n$ are said to be mutually exclusive if the occurrence of anyone of them implies the non-occurrence of the remaining $n − 1$ events.

So in your example, the three events are definitely not mutually exclusive (if 3 or 4 happened, 4 or 5 could have happened). If you altered your events to this: Roll 1, Roll 3, Roll 4 or 5, they would be mutually exclusive.

Does that answer your question?

 

[Spud]

According to the wiki article:

So in your example, the three events are definitely not mutually exclusive (if 3 or 4 happened, 4 or 5 could have happened). If you altered your events to this: Roll 1, Roll 3, Roll 4 or 5, they would be mutually exclusive.

Does that answer your question?

Hey, thanks for helping out.

So, yes, it does answer my question ... sort of.

Let's assume that altering the events is not an option, so we're stuck with:

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

I guess I'm still confused because depending on which definition we use, the three events can either be mutually exclusive or not mutually exclusive.

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

With this definition, the three events above are not mutually exclusive.

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

With this definition, the three events above are mutually exclusive.

I have checked countless sources and there are many, many different definitions, but the two most common definitions I have found seem to be some kind of variant of the above two definitions.

So I guess my question is: Which definition is correct and why?

Cheers!

 

[Ackbach]

Hey, thanks for helping out.

So, yes, it does answer my question ... sort of.

Let's assume that altering the events is not an option, so we're stuck with:

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

I guess I'm still confused because depending on which definition we use, the three events can either be mutually exclusive or not mutually exclusive.

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

This is correct.

With this definition, the three events above are not mutually exclusive.

This is correct.

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

This is definitely incorrect. It should read like this:

A set of events is mutually exclusive if the probability of any two distinct events in the set occurring at the same time is $0.$

 

[Spud]

This is definitely incorrect. It should read like this:

A set of events is mutually exclusive if the probability of any two distinct events in the set occurring at the same time is $0.$

Correct me if I'm wrong, but I think you're mistaken with this definition you've provided. Basically, I don't think it makes sense.

Allow me to explain. Our events are:

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

Therefore, the set of events is {[1], [3, 4], [4, 5]} ... right? Or am I wrong here?

So, if we are trying to determine if a set of events (which, in this case, includes three events; no more and no less) are mutually exclusive, then we need to consider all three events ... don't we? It just doesn't seem to make sense if we pick two of the three events, and then classify the set of three events as mutually exclusive or not mutually exclusive based on two of the events.

I hope I sufficiently explained my concerns here. Of course, I am the one that is probably mistaken, but I don't really understand why.

Again, thanks for your help!

 

[Ackbach]

Correct me if I'm wrong, but I think you're mistaken with this definition you've provided. Basically, I don't think it makes sense.

Allow me to explain. Our events are:

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

Therefore, the set of events is {[1], [3, 4], [4, 5]} ... right? Or am I wrong here?

So, if we are trying to determine if a set of events (which, in this case, includes three events; no more and no less) are mutually exclusive, then we need to consider all three events ... don't we?

Yes, and this definition does that. But it does it as follows:

Step 1. Compare Event 1 to Event 2: mutually exclusive.
Step 2. Compare Event 1 to Event 3: mutually exclusive.
Step 3. Compare Event 2 to Event 3: not mutually exclusive.

Therefore the set is not mutually exclusive.

While you're only comparing two events at a time, you perform all possible such comparisons. By the time you've done that, every event will have been compared to every other event. Only after you've finished doing that can you say whether the set as a whole is mutually exclusive or not.

It just doesn't seem to make sense if we pick two of the three events, and then classify the set of three events as mutually exclusive or not mutually exclusive based on two of the events.

As I hope I've shown, that's not what we're doing.

The key word in the definition I provided is "any". That means you have to perform every possible comparison.

 

[Spud]

Yes, and this definition does that. But it does it as follows:

Step 1. Compare Event 1 to Event 2: mutually exclusive.
Step 2. Compare Event 1 to Event 3: mutually exclusive.
Step 3. Compare Event 2 to Event 3: not mutually exclusive.

Therefore the set is not mutually exclusive.

While you're only comparing two events at a time, you perform all possible such comparisons. By the time you've done that, every event will have been compared to every other event. Only after you've finished doing that can you say whether the set as a whole is mutually exclusive or not.

The keyword in the definition I provided is "any". That means you have to perform every possible comparison.

Ahhh, this makes much more sense to me now; thank you so much!

I think I'm in a position where I now completely agree with you that the three events are not mutually exclusive.

But now I have a follow-up question, if you don't mind?

If the three events are not mutually exclusive, is that synonymous with saying that the three events are mutually inclusive?

My understanding of mutual inclusivity:
Two events are mutually inclusive if it's possible for those two events to occur at the same time.

I'm not sure how to define it, though, if we're talking about a set of events that consists of more than two events.

Possible definition 1:
A set of events are mutually inclusive if it’s possible for all of the events in that set to occur at the same time.

Possible definition 2:
A set of events are mutually inclusive if the occurrence of each event does not preclude the occurrence of all the other events in that set.

But I'm not sure if either of these definitions are coherent. What are your thoughts?

Anyway, so if we go through the steps:

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

Step 1: Compare Event 1 to Event 2: Mutually exclusive.
Step 2: Compare Event 1 to Event 3: Mutually exclusive.
Step 3: Compare Event 2 to Event 3: Mutually inclusive.

So if we agree that the three events are not mutually exclusive, does that mean they are mutually inclusive? If not, then that means that the three events are not mutually exclusive and not mutually inclusive. If this is the case, is there a specific term to describe this scenario? I cannot stop thinking about this!

Thanks so much for all your help!

 

[Ackbach]

Sure, you can define that term if you like. One way to describe the events is that they have non-empty intersection (if you want it possible for all of them to occur simultaneously).

 

[Spud]

Sure, you can define that term if you like. One way to describe the events is that they have non-empty intersection (if you want it possible for all of them to occur simultaneously).

Sorry, but you are saying that you consider the three events (Roll 1, Roll 3 or 4, Roll 4 or 5) to be mutually inclusive?

 

[Ackbach]

Sorry, but you are saying that you consider the three events (Roll 1, Roll 3 or 4, Roll 4 or 5) to be mutually inclusive?

I would actually define the term like this (it's not a standard term, so anything goes!):

A set of events is mutually inclusive if the occurrence of anyone event in the set does not preclude the occurrence of any other event in the set.

This would parallel the mutually exclusive definition as being pairwise defined.

With this definition, the events you have there are not mutually inclusive. Indeed, I don't think your set of events would be mutually inclusive for any of these three definitions.

 

[Spud]

I would actually define the term like this (it's not a standard term, so anything goes!):

A set of events is mutually inclusive if the occurrence of anyone event in the set does not preclude the occurrence of any other event in the set.

This would parallel the mutually exclusive definition as being pairwise defined.

With this definition, the events you have there are not mutually inclusive. Indeed, I don't think your set of events would be mutually inclusive for any of these three definitions.

Hey Ackbach,

Thanks for your response and clarification.

I have been thinking about this and I agree with everything you've said.

But I am still confused about what term we should use to classify this situation?

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

This set of three events are not mutually exclusive.
This set of three events are not mutually inclusive.

So, what are they then?

Is there a term for a set of events that are neither mutually exclusive nor mutually inclusive?

Prior to discussing ME and MI on various forums, I was under the impression that ME and MI were opposites. It now seems that the impression of ME and MI being opposites is not true?

Cheers!

 

[Ackbach]

Right, I would agree that they are not opposites, since a third option exists. It's no different from open and closed intervals. $(0,1)$ is an open interval, and $[0,1]$ is a closed interval. But $(0,1]$ is neither. This sometimes happens when we can't define opposites, or we can't exclude other options.

 

[Spud]

Right, I would agree that they are not opposites, since a third option exists. It's no different from open and closed intervals. $(0,1)$ is an open interval, and $[0,1]$ is a closed interval. But $(0,1]$ is neither. This sometimes happens when we can't define opposites, or we can't exclude other options.

Okay great; thanks again for all your help!

 

[Ackbach]

Okay great; thanks again for all your help!

You're welcome!