Let A and B be two events such that P(A \(\cup\) B)’ = 1/6, P(A \(\cap\) B) = 1/4 and P(A)’ = 1/4 where A’ stands for complement of A. Then prove that events A and B independent

Solution :

Given P(A \(\cup\) B)’ = 1/6, P(A \(\cap\) B) = 1/4 and P(A)’ = 1/4

\(\therefore\)   P(A \(\cup\) B) = 1 – P(A \(\cup\) B)’

= 1 – \(1\over 6\) = \(5\over 6\)

and P(A) = 1 – P(A)’ = 1 – \(1\over 4\) = \(3\over 4\)

P(A \(\cup\) B) = P(A) + P(B) – P(A \(\cap\) B)

\(\implies\) \(5\over 6\) = \(3\over 4\) + P(B) – \(1\over 4\)

P(B) = \(1\over 3\)

\(\implies\) A and B are not equally likely,

Also, P(A \(\cap\) B) = P(A).P(B) = \(1\over 4\)

So, events are independent.

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