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Question
Events A and Bare such that P(A) = `1/2`, P(B) = `7/12` and `P(barA ∪ barB) = 1/4`. Find whether the events A and B are independent or not.
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Solution
Given, P(A) = `1/2`, P(B) = `7/12`
And `P(barA ∪ barB) = 1/4`
For A and B are independent
P(A ∩ B) = P(A).P(B) ...(i)
Now, `P(barA ∪ barB)` = P(A ∩ B)
⇒ `P(barA ∪ barB)` = 1 – P(A ∩ B)
⇒ P(A ∩ B) = `1 - P(barA ∪ barB)`
⇒ P(A ∩ B) = `1 - 1/4 = 3/4` ...(ii)
Now, P(A).P(B) = `1/2 xx 7/12 = 7/24` ...(iii)
Since from equations (ii) and (ill)
P(A ∩ B) ≠ P(A).P(B)
Therefore, events A and B are not independent.
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