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Question
Solve the following:
If P(A) = `"P"("A"/"B") = 1/5, "P"("B"/"A") = 1/3` the find `"P"("B'"/"A'")`
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Solution
It is given that, P(A) = `"P"("A"/"B") = 1/5`
`"P"("B"/"A") = 1/3`
Now P(A ∩ B) = `"P"("A")*"P"("B"/"A") = 1/5*1/3 = 1/15`
Also, P(A ∩ B) = `"P"("B")*"P"("A"/"B")`
∴ `1/15 = "P"("B")*1/5`
∴ P(B) = `1/3`
∴ P(A)·P(B) = `1/5*1/3 = 1/15` = P(A ∩ B)
∴ A, B are independent
∴ A', B; A', B' are also independent
`"P"("B'"/"A'") = ("P"("B'" ∩ "A'"))/("P"("A'"))`
= `("P"("B'")*"P"("A'"))/("P"("A'"))` ...[∵ A' and B' are independent]
= 1 – P(B)
= `1 - 1/3`
= `2/3`
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