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Question
If P(A) = `2/5`, P(B) = `3/10` and P(A ∩ B) = `1/5`, then P(A|B).P(B'|A') is equal to ______.
Options
`5/6`
`5/7`
`25/42`
1
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Solution
If P(A) = `2/5`, P(B) = `3/10` and P(A ∩ B) = `1/5`, then P(A|B).P(B'|A') is equal to `25/42`.
Explanation:
Given that: P(A) = `2/5`, P(B) = `3/10` and P(A ∩ B) = `1/5`
P(A') = `1 - 2/5 = 3/5`
P(B') = `1 - 3/10 = 7/10`
And P(A' ∩ B') = 1 – P(A ∪ B)
= 1 – [P(A) + P(B) – P(A ∩ B)]
= `1 - [2/5 + 3/10 - 1/5]`
= `1 - [1/5 + 3/10]`
= `1 - 5/10`
= `1/2`
∴ `"P"("A'"/"B'") = ("P"("A'" ∩ "B'"))/("P"("B'"))`
= `(1/2)/(7/10)`
= `5/7`
And `"P"("B'"/"A'") = ("P"("A'" ∩ "B'"))/("P"("A'"))`
= `(1/2)/(3/5)`
= `5/6`
∴ `"P"("A'"/"B'")*"P"("B'"/"A'") = 5/7 xx 5/6`
= `25/42`
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