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Question
If A and B are two events such that P(A|B) = p, P(A) = p, P(B) = `1/3` and P(A ∪ B) = `5/9`, then p = ______.
Solution
If A and B are two events such that P(A|B) = p, P(A) = p, P(B) = `1/3` and P(A ∪ B) = `5/9`, then p = `1/3`.
Explanation:
Given that, P(A) = p
P(B) = `1/3`
And P(A ∪ B) = `5/9`
P(A|B) = `("P"("A" ∩ "B"))/("P"("B"))` = p
⇒ P(A ∩ B) = p
P(B) = `"p"*1/3` and P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
`5/9 = "p" + 1/3 - "p"/3`
⇒ `5/9 - 1/3 = (2"p")/3`
⇒ `2/9 = (2"p")/3`
⇒ p = `1/3`
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