# Probability of an Event

## Probability of Class 12

## Probability of an Event

In performance of a random experiment the occurrence of any event is always uncertain but a measure of its probable occurrence can be devised. Which then is called probability of the event. To devise this measure following conventions are made:

(i) Probability of null event is 0

(ii) Probability of sure event is 1

(iii) Probability of an event which is less likely to occur is smaller than the probability of the event which is more likely to occur.

### Classical Approach

The sample space is assumed to be finite, its sample points are equally likely, pair-wise mutually exclusive and taken together form an exhaustive system.

Let S denotes the sample space of a random experiment and A be an event. Then the probability of A is defined by the rational number

P(A) = =

### Some Theorems

1. If A ⊂ B then (i) P(A) ≤ P(B) and (ii) P(B − A) = P(B) −P(A).

2. P(Φ) = 0

3. P(S) = 1

4. 0 ≤ P(A) ≤ 1

5. P(A′) = 1 − P(A)

6. P(B − A) = P(B − (A ∩ B)) = P(B) − P(A ∩ B).

7. P(A ∪ B) = P(A) + P(B) −P(A∩B)

8. P(A ∪ B) = P(A) + P(B) if A ∩ B = Φ.

9. P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) − P(B ∩ C) − P(C ∩ A) + P(A ∩ B ∩ C).

10. P(A) = P(A ∩ B) + P(A ∩ B′).

11. If A ⊂ A_{1} ∪ A_{2} ∪ A_{3} ∪…∪ An and A ∩ A_{1}, A ∩ A_{2}, A ∩ A_{3} … and A ∩ An are pair-wise mutually exclusive, then P(A) = P(A ∩ A1) + P(A ∩ A_{2}) + P(A ∩ A_{3}) + … + P(A ∩ A_{n}).