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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. |
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Answer» 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. |
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