In a random experiment, let S be the sample space. Let A ⊆ S and B ⊆ S be the events in S.

**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*.

Tags : Probability Probability

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