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Question
Solve the following:
Let A and B be independent events with P(A) = `1/4`, and P(A ∪ B) = 2P(B) – P(A). Find `"P"("B'"/"A")`
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Solution
A and B are independent events.
∴ P(A ∩ B) = P(A) × P(B)
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
∴ P(A ∪ B) = P(A) + P(B) – P(A) × P(B)
∴ 2P(B) – P(A) = P(A) + P(B) – P(A) × P(B) ...[∵ P(A ∪ B) = 2P(B) – P(A)]
∴ `2"P"("B") - 1/4 = 1/4 + "P"("B") - 1/4 xx "P"("B")`
∴ `2"P"("B") - "P"("B") + 1/4 "P"("B") = 1/4 + 1/4`
∴ `5/4 "P"("B") = 2/4`
∴ P(B) = `2/5`
`"P"("B'"/"A") = ("P"("B'" ∩ "A"))/("P"("A"))`
= `("P"("B'") xx "P"("A"))/("P"("A"))`
= P(B')
= 1 – P(B)
= `1 - 2/5`
= `3/5`
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