Answer
a)The event “none of the three events occurs” is represented as \[({{A}^{C}}\cap {{B}^{C}}\cap {{C}^{C}})\].
b)The event “all three of the events occur” is represented as \[(A\cap B\cap C)\].
c)The event “only event A occurs” is representing as\[(A\cap {{B}^{C}}\cap {{C}^{C}})\].
d)The event “exactly one event occurs” is represented as:.
\[(A\cap {{B}^{C}}\cap {{C}^{C}})\cup ({{A}^{C}}\cap B\cap {{C}^{C}})\cup ({{A}^{C}}\cap {{B}^{C}}\cap C)\]
e)
The event “exactly two events occur” is represented as:
\[(A\cap B\cap {{C}^{C}})\cup ({{A}^{C}}\cap B\cap C)\cup (A\cap {{B}^{C}}\cap C)\]
Work Step by Step
(a)
Let A, B and C be any three events defined on a sample space S.
The complement of the events A, B and C are \[{{A}^{C}},\text{ }{{B}^{C}}\text{ and }{{C}^{C}}\]respectively.
Since, the events \[{{A}^{C}}\]means event A does not occur.
So, “none of the three events occurs,” is equivalent to A does not occur and B does not occur and C does not occur.
Therefore, these can be written in symbolic form as: \[({{A}^{C}}\cap {{B}^{C}}\cap {{C}^{C}})\]
(b)
Let A, B and C be any three events defined on a sample space S.
So, “all three of the events occur,” means the intersection of all three events.
Therefore, these can be written in symbol form as: \[(A\cap B\cap C)\]
(c)
Let A, B and C be any three events defined on a sample space S.
The complement of the events A, B and C are \[{{A}^{C}},\text{ }{{B}^{C}}\text{ and }{{C}^{C}}\]respectively.
Since, the events \[{{A}^{C}}\]means event A does not occur.
So, “only event A occurs,” is equivalent to A occur and B does not occur and C does not occur.
Therefore, these can be written in symbol form as: \[(A\cap {{B}^{C}}\cap {{C}^{C}})\]
(d)
Let A, B and C be any three events defined on a sample space S.
The complement of the events A, B and C are \[{{A}^{C}},\text{ }{{B}^{C}}\text{ and }{{C}^{C}}\]respectively.
The events \[{{A}^{C}}\]means event A does not occur.
So, “exactly one event occurs,” is equivalent to events A or B or C occur but not all 3 events occur or any two events occur.
Therefore, these can be written in symbol as:
\[(A\cap {{B}^{C}}\cap {{C}^{C}})\cup ({{A}^{C}}\cap B\cap {{C}^{C}})\cup ({{A}^{C}}\cap {{B}^{C}}\cap C)\]
(e)
Let A, B and C be any three events defined on a sample space S.
The complement of the events A, B and C are \[{{A}^{C}},\text{ }{{B}^{C}}\text{ and }{{C}^{C}}\]respectively.
Since, the events \[{{A}^{C}}\]means event A does not occur.
So, “exactly two events occur,” is equivalent to events A or B occur but C not occurs and B or C occur but A not occurs and A or C occur but B not occurs.
Therefore, these can be written in symbol form as:
\[(A\cap B\cap {{C}^{C}})\cup ({{A}^{C}}\cap B\cap C)\cup (A\cap {{B}^{C}}\cap C)\]
