Class 12 Maths Notes: Probability – Mutual and Pairwise Independence

Mutual Independence and Pairwise Independence:

Three events A, B, C are said to be mutually independent if,

P(A∩B) = P(A)×P(B), P(A ∩ C) = P(A)×P(C), P(B ∩ C) = P(B)×P(C)

and P(A∩B∩C) = P(A)×P(B)×P(C)

These events would be said to be pairwise independent if,

P(A∩B) = P(A)×P(B), P(B∩C) = P(B)×P(C) and P(A∩C) = P(A)×P(C).

Thus mutually independent events are pairwise independent but the converse may not be true.

Example -: An event A₁ can happen with probability p₁ and event A₂ can happen with probability p₂.

What is the probability that

(i) exactly one of them happens

(ii) at least one of them happens(Given A₁ and A₂ are independent events).

Sol:                  (i) The probability that A₁ happens is p₁

∴ The probability that A₁ fails is 1 – p₁

Also the probability that A₂ happens is p₂

Now, the chance that A₁ happens and A₂ fails is p₁(1 – p₂) and the chance that A₁ fails and A₂ happens is p₂(1-p₁)

∴ The probability that one and only one of them happens is

p₁(1 – p₂) + p₂(1 – p₁) = p₁ + p₂ – 2p₁p₂

(ii) The probability that both of them fail = (1 – p₁) (1 – p₂).

∴ probability that atleast one happens=1 -(1 – p₁) (1 – p₂)=p₁+p₂ – p₁ p₂

<All Probability Topics Maths Notes Physics Notes Chemistry Notes Biology Notes