(a) For Two Events. If A and B are two events associated with a random experiment, then
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
⇒ P(A or B) = P(A) + P(B) — P(A and B)
Corollary 1: If A and B are mutually exclusive events, then,
P(A ∩ B) = 0, therefore
P(A ∪ B) = P(A) + P(B)
Corollary 2: P(A or B) ≤ P((\underline{A})) + P(B)
Given, P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Since, P(A ∩ B) is greater than or equal to 0,
P(A or B) < P(A) + P(B)
Equality in the above result holds when A and B are mutually exclusive as P(A ∩ B) = 0
(b) For three events
If A, B and C be any three events associated with a random experiment, then
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(A ∩ C) – P(B ∩ C) + P(A ∩ B ∩ C)
Corollary 1: If A, B and C are mutually exclusive events, then
P(A ∩ B) = P(B ∩ C) = P(A ∩ C) = P(A ∩ B ∩ C) = 0
∴ P(A ∪ B ∪ C) = P(A) + P(B) + P(C)
Note : (i) If A and B are two events such that A ⊆ B, then P(A) ≤ P(B)
(ii) If E is an event associated with a random experiment, then 0 P(E) ≤ 1
