# Confused About Mutual Exclusivity with More Than Two Events

#### Muppet

[FONT=&amp]Hello,[/FONT]

[FONT=&amp]Sorry if this is the wrong section of the forums, but I figured that questions about mutually exclusive events are relevant to probability.[/FONT]

[FONT=&amp]My current understanding:[/FONT]
[FONT=&amp]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.[/FONT]

[FONT=&amp]I guess I'll use an example with fair six-sided dice to try and explain where my confusion lies.[/FONT]

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

[FONT=&amp]My question:[/FONT]
[FONT=&amp]Are the three events [/FONT][FONT=&amp]Roll 1[/FONT][FONT=&amp], [/FONT][FONT=&amp]Roll 3 or 4[/FONT][FONT=&amp] and [/FONT][FONT=&amp]Roll 4 or 5[/FONT][FONT=&amp] considered to be [/FONT][FONT=&amp]mutually exclusive[/FONT][FONT=&amp] or [/FONT][FONT=&amp]not mutually exclusive[/FONT][FONT=&amp]?[/FONT]

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

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

[FONT=&amp]Thanks a lot![/FONT]

#### romsek

MHF Helper
Mutually exclusive means the probability of any and all subsets of the events happening simultaneously is zero.

In your example the 3 events are not mutually exclusive as a roll of 4 is a member of the second and third events.

The fact that event 1 is mutually exclusive with both events 2 and 3 doesn't change this.

#### Muppet

Mutually exclusive means the probability of any and all subsets of the events happening simultaneously is zero.

In your example the 3 events are not mutually exclusive as a roll of 4 is a member of the second and third events.

The fact that event 1 is mutually exclusive with both events 2 and 3 doesn't change this.
Thanks for the response.

Interesting. In your definition, I'm wondering why you included the condition any? I've never seen that before; I've only ever seen the condition all. But you've got both.

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.

Your definition is similar to the second definition, but you've added the any condition.

The way that I'm thinking about it is that if we are talking about all three events, then we must consider all three events together. So if all three events cannot occur at the same time, those three events are mutually exclusive. And if all three events can occur at the same time, those three events are mutually inclusive.

Hopefully, I've sufficiently explained my thought process here. I get a feeling that I'm still wrong, though, so could you please help me understand why? What are your thoughts about the two definitions I've provided above? Which one is correct (if any) and why?

Thanks!

#### Muppet

Mutually exclusive means the probability of any and all subsets of the events happening simultaneously is zero.

In your example the 3 events are not mutually exclusive as a roll of 4 is a member of the second and third events.

The fact that event 1 is mutually exclusive with both events 2 and 3 doesn't change this.
Thank you for helping me understand; it's much appreciated!

As a side question, would the events then be considered mutually inclusive?