Question 1199933: Suppose one die is rolled. Consider the following events:
A : the die comes up even
B : the die comes up odd
C : the die comes up 4 or more
D : the die comes up at most 2
E : the die comes up 3
a) Are events C and E mutually exclusive?
b) Are events B and C mutually exclusive?
c) Are events C, D and E mutually exclusive?
d) Are there any four mutually exclusive events among A, B, C, D and E?
e) Are events A and B mutually exclusive?
Answer by math_tutor2020(3816)   (Show Source):
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a) Yes, the events C and E are mutually exclusive.
It is impossible for both events C and E to occur at the same time. In other words, set C and set E have nothing in common.
C = {4,5,6}
E = {3}
This is what it means to be mutually exclusive. Another term is "disjoint".
If event C happens, then the die shows 4, 5, or 6.
It's fairly clear that event E cannot happen since we don't have 3 show up. The same works in reverse as well.
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b) No, the events B and C are not mutually exclusive.
B = {1,3,5}
C = {4,5,6}
The overlapped or shared element is '5'
It's possible for events B and C to occur simultaneously. This is why B and C are not mutually exclusive.
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c) Yes, events C, D, and E are mutually exclusive.
C = {4,5,6}
D = {1,2}
E = {3}
The three sets have nothing in common.
They partition the sample space {1,2,3,4,5,6} which is the set of possible outcomes of a die roll.
C union D union E = {4,5,6} union {1,2} union {3} = {1,2,3,4,5,6}
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d) No, there aren't any collection of four sets that are mutually exclusive.
There are 5 sets total to represent the 5 events.
A = {2,4,6}
B = {1,3,5}
C = {4,5,6}
D = {1,2}
E = {3}
We're tasked to find 4 sets that are mutually exclusive. We must kick out one of those five sets.
Let's say we erased set A to be left with
B = {1,3,5}
C = {4,5,6}
D = {1,2}
E = {3}
Earlier we found that events C,D,E were mutually exclusive, and they partition the entire sample space.
Since these three sets union to form the entire sample space, there isn't any room for another set to be mutually exclusive to these three.
The introduction of set B breaks the "mutual exclusivity".
This means "C,D,E" (in any order) will not be part of a group of 4 mutually exclusive sets.
Erasing set B will lead to the same conclusion as the previous paragraph.
Let's say we erased set C
A = {2,4,6}
B = {1,3,5}
D = {1,2}
E = {3}
But now we run into the problem of sets A and B unioning together to form the entire sample space {1,2,3,4,5,6}
Introducing set D or E will break the "mutual exclusivity".
Erasing set D will lead to the same logic and conclusion as the previous paragraph.
Erasing set E will lead to the same logic and conclusion as the previous paragraph.
This is why there aren't any collection of 4 sets that are not mutually exclusive.
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e) Yes, A and B are mutually exclusive.
A = {2,4,6}
B = {1,3,5}
The two sets have nothing in common.
If a number is even, then it cannot be odd, and vice versa.
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