Events A and B are such that P(A) = 1/2​, P(B) = 7/12 and = P(A' B') = 1/4​. Find whether the events A and B are independent or not. ​

Question

Events A and B are such that P(A) = \(\frac12\)​, P(B) = \(\frac7{12}\) and \(P( A'\cup B')\) = \(\frac14\)​. 

Find whether the events A and B are independent or not.

Answer 1

It is given that  P(A) = \(\frac12\)​, P(B) = \(\frac7{12}\) and \(P( A'\cup B')\) = \(\frac14\)​. 

⇒ \(P( A'\cup B')= \frac14\) 

⇒​ \(P((A\cap B)')= \frac14\)      \([A'\cup B' = (A\cap B)']\)

⇒ \(1-P(A\cap B)= \frac14\)

⇒ \(P(A\cap B)= \frac34\)  ......(1)

However, P(A) ⋅ P(B) = \(\frac 12.\frac7{12} = \frac7{24}\) .....(2)

Here, \(\frac34 \ne \frac7{24}\)

∴ P(A ∩ B) \(\ne\) P(A) ⋅ P(B)

Therefore, A and B are not independent events.