Algebra of Events

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.