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Question
Events A and B are such that `P(A) = 1/2, P(B) = 7/12 and P("not A or not B") = 1/4` . State whether A and B are independent?
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Solution
P(A - not and B - not) = `P (barA cap barB) = P(barA cup barB) = 1 - P (Acup B)`
= 1 − [P(A) + P(B) − P(A ∩ B)]
⇒ `1/4 = 1 - [1/2 + 7/12 - P(A ∩ B)]`
⇒ P(A ∩ B) = `1/4 - 1 + 1/2`
= `(3 - 12 + 6 + 7)/12`
= `4/12`
= `1/3`
and P(A) . P(B) = `1/2 . 7/12`
= `7/24 ≠ P(A ∩ B)`
⇒ P (A) × P (B) ≠ P (A ∩ B)
Events A and B are not independent.
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