Independent Events

Probability of Class 12

Independent Events

Two events are said to be independent if the probable occurrence or non-occurrence of  any one is not affected by the occurrence or non-occurrence of the other. i.e. two events A and B are independent if

P(A/B) = P(A/B′) = P(A) or P (B/A) = P(B/A′) = P(B) or P(A ∩ B) = P(A).P(B).

  • Three events A, B and C are independent if P(A∩B) = P(A).P(B), P(B ∩C) = P(B).P(C), P(C∩A) = P(C).P(A) and  P(A ∩ B ∩ C) = P(A). P(B). P(C).
  • Three events A, B and C are called pair-wise independent if P(A ∩ B) = P(A). P(B),

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


Three independent events are always pair-wise independent, but its converse may not be true.

Some Results

If A and B are independent, then

(i) A and B′ are independent.

(ii) A′ and B are independent.

(iii) A′ and B′ are independent.

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