Probability of an Event

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.

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 ⊂ A₁ ∪ A₂ ∪ A₃ ∪…∪ An and A ∩ A₁, A ∩ A₂, A ∩ A₃ … and A ∩ An are pair-wise mutually exclusive, then P(A) = P(A ∩ A1) + P(A ∩ A₂) + P(A ∩ A₃) + … + P(A ∩ A_n).