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October 4th, 2019, 06:19 AM   #1
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Confused About Mutual Exclusivity with More Than Two Events

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!
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October 4th, 2019, 02:40 PM   #2
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Quote:
Originally Posted by Spud View Post
. . . two events are mutually exclusive if the probability of both events occurring at the same time is 0.
For three or more events, they are mutually exclusive iff the probability of any two of them occurring at the same time is zero. In the example you gave, the three events aren't mutually exclusive.

More accurately, three or more events are mutually exclusive iff no pair of them can occur simultaneously.

Last edited by skipjack; October 5th, 2019 at 06:56 AM.
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October 4th, 2019, 03:56 PM   #3
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When talking about more than two events, there seems to be some ambiguity in the definition. All pairs are mutually exclusive (?), or one event is exclusive from any other?
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October 5th, 2019, 04:38 AM   #4
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Only Roll 1 and Roll 3 or 4 mutually exclusive.
And we do not know whether Roll 1 and Roll 4 or 5 is mutually exclusive or not.

Now, the probability of all three events is zero. But that does not mean all three events are mutually exclusive.

Two events are mutually exclusive if they both cannot occur at the same time. And these events are dependent events. One outcome is dependent on the other one.
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October 5th, 2019, 10:46 AM   #5
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Quote:
Originally Posted by skipjack View Post
For three or more events, they are mutually exclusive iff the probability of any two of them occurring at the same time is zero. In the example you gave, the three events aren't mutually exclusive.

More accurately, three or more events are mutually exclusive iff no pair of them can occur simultaneously.
Okay, I think I see what you're saying.

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

Correct me if I misunderstood, but even though Event 1 and Event 2 are mutually exclusive, and Event 1 and Event 3 are mutually exclusive, the set of Event 1, Event 2 and Event 3 are not mutually exclusive simply because Event 2 and Event 3 are not mutually exclusive.

Is this what you're saying?
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October 5th, 2019, 10:51 AM   #6
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Quote:
Originally Posted by mathman View Post
When talking about more than two events, there seems to be some ambiguity in the definition. All pairs are mutually exclusive (?), or one event is exclusive from any other?
Hmm.

I am specifically asking about the set of three events. But as you said, there seems to be a couple of working definitions, that when applied to more than two events, produce different results.

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

I could be wrong, but based on some feedback from others, I think the following is correct:

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.

Because Event 2 and Event 3 are not mutually exclusive, the set of Event 1, Event 2 and Event 3 are not mutually exclusive.

What do you think?
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October 5th, 2019, 10:57 AM   #7
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Quote:
Originally Posted by tahirimanov19 View Post
Only Roll 1 and Roll 3 or 4 mutually exclusive.
And we do not know whether Roll 1 and Roll 4 or 5 is mutually exclusive or not.

Now, the probability of all three events is zero. But that does not mean all three events are mutually exclusive.

Two events are mutually exclusive if they both cannot occur at the same time. And these events are dependent events. One outcome is dependent on the other one.
I'm not sure that I agree with you. I think we do know if Roll 1 and Roll 4 or 5 are mutually exclusive or not, but please let me know if you think I'm wrong.

I understand the criteria for a set of two events to be mutually exclusive, but I'm specifically asking about a set of three (or more) events.

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.

Based on what others have said, it seems to follow that because Event 2 and Event 3 are not mutually exclusive, the set of Event 1, Event 2 and Event 3 are not mutually exclusive.

What do you think?
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October 5th, 2019, 03:56 PM   #8
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Quote:
Originally Posted by Spud View Post
. . . even though Event 1 and Event 2 are mutually exclusive, and Event 1 and Event 3 are mutually exclusive, the set of Event 1, Event 2 and Event 3 are not mutually exclusive simply because Event 2 and Event 3 are not mutually exclusive.

Is this what you're saying?
Yes.
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October 5th, 2019, 04:19 PM   #9
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I am going to use terms $E_1, \; E_2, \; E_3$ for three events.
(i) $E_1 \; \& \; E_2 $ are M.E.
(ii)$E_1 \; \& \; E_3 $ are M.E.
(iii)$E_2 \; \& \; E_3 $ are not M.E.

Now, take a paper and draw a circle, and write $E_1$ in it.
Draw another circle which does not intersect with $E_1$, and write $E_2$ in it.
Draw the third circle which does not intersect with the first ($E_1$) circle, but intersects with the second (E_2) circle, and write $E_3$ in it.

Now, non-intersecting circles represent mutually exclusive events, which by default dependent events, meaning one outcome affects the other(s).
$P(A \& B) =0$ if A and B are M.E.
Intersecting circles represent independent events, meaning you can both outcomes, one does not affect the other.

Now the set of all three events is M.E., because of (i) and (ii).
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October 5th, 2019, 11:56 PM   #10
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Three or more events are called mutually exclusive, or disjoint, if each pair of events is mutually exclusive. That's the exact wording from a textbook. Have you found another textbook that disagrees with that?
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