Algebra
of Events
In a random experiment,
let S be the sample space. Let A ⊆ S and B ⊆
S be the events in S. We say that
(i) (A Ո B) is an event that occurs only when both A
and B occurs.
(ii) (A U B) is an event
that occurs when either one of A or B occurs.
(iii) is
an event that occurs only when A doesn’t occur.
Note:
·
A ∩ = ɸ
·
A∪ = S
·
If A, B are mutually exclusive events,
then P (A ∪ B) = P(A) + P (B)
·
P (Union of mutually exclusive events)
= ∑( Probability of events)
Thorem
1
If A and B
are two events associated with a random experiment, then prove that
(i) P (A Ո ) = P (only A) = P (A) −P (A ∩ B)
(ii)
P ( Ո B) = P(only B) = P (B ) −P (A
∩ B)
Proof
(i) By Distributive
property of sets,
Therefore, the events A
Ո B and A Ո are mutually exclusive whose union is A.
(ii) By Distributive
property of sets,
Therefore, the events A
Ո B and Ո B are mutually exclusive whose union is B.
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