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Question
Solve the following:
If P(A ∩ B) = `1/2`, P(B ∩ C) = `1/3`, P(C ∩ A) = `1/6` then find P(A), P(B) and P(C), If A,B,C are independent events.
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Solution
Since A and B are independent events,
P(A ∩ B) = P(A) · P(B)
∴ P(A) P(B) = `1/2` ...(i)
B and C are independent events.
∴ P(B ∩ C) = P(B) · P(C)
∴ P(B) P(C) = `1/3` ...(ii)
A and C are independent events.
∴ P(A ∩ C) = P(A) · P(C)
∴ P(A) P(C) = `1/6` ...(iii)
Dividing (i) by (ii), we get
`("P"("A") * "P"("B"))/("P"("B") * "P"("C")) = (1/2)/(1/3)`
∴ P(A) = `3/2` P(C) ...(iv)
Substituting equation (iv) in (iii), we get
`3/2`P(C) · P(C) = `1/6`
∴ [P(C)]2 = `1/9`
∴ P(C) = `1/3`
Substituting P(C) = `1/3` in equation (ii), we get P(B) = 1
Substituting P(B) = 1 in equation (i), we get P(A) = `1/2`
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