For two events if A and B are independent then prove that A', B' are also independent.

Engineering

Mathematics

Conditional Probability

Question

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

Given A, B are independent then P(A ⋂ B) = P(A)P(B).......(1).

Now

P(A' ⋂ B') = P{ (A ⋃ B)'} [ Using De Morgan's Law]

or, P(A' ⋂ B') = 1 – P(A ⋃ B)

or, P(A' ⋂ B') = 1 – {P(A) + P(B) – P(A ⋂ B)}

or, P(A' ⋂ B') = 1 – {P(A) + P(B) – P(A)P(B)} [ Using (1)]

or, P(A' ⋂ B') = 1 – P(A) – P(B) + P(A)P(B)

or, P(A' ⋂ B') = (1 – P(A))(1 – P(B))

or, P(A' ⋂ B') = P(A')P(B')

This given A' B' are independent.

Please subscribe our Youtube channel to unlock this solution.