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Question
Solve the following:
For three events A, B and C, we know that A and C are independent, B and C are independent, A and B are disjoint, P(A ∪ C) = `2/3`, P(B ∪ C) = `3/4`, P(A ∪ B ∪ C) = `11/12`. Find P(A), P(B) and P(C)
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Solution
It is given that
P(A ∪ C) = `2/3`, P(B ∪ C) = `3/4`, P(A ∪ B ∪ C) = `11/12`.
P(A ∪ C) = `2/3` gives,
P(A) + P(C) – P(A ∩ C) = `2/3` ...(1)
P(B ∪ C) = `3/4` gives,
P(B) + P(C) – P(B ∩ C) = `3/4` ...(2)
P(A ∪ B ∪ C) = `11/12` gives,
P(A) + P(B) + P(C) – P(A ∩ B) – P(A ∩ C) – P(B ∩ C) + P(A ∩ B ∩ C) = `11/12`
∴ P(A) + P(B) + P(C) – P(A n C) – P(B n C) = `11/12 ...[(because "A""," "B" "are disjoint"),(therefore "A" ∩ "B" = "A" ∩ "B" ∩ "C" = phi)]`
∴ `"P"("A") + "P"("B") + "P"("C") – ["P"("A") + "P"("C") - 2/3] - ["P"("B") + "P"("C") - 3/4] = 11/12` ...[By (1) and (2)]
∴ −P(C) = `11/12 - 2/3 - 3/4 = -1/2`
∴ P(C) = `1/2`
From (1),
P(A) + P(C) – P(A)·P(C) = `2/3` ...[∵ A, C are independent]
∴ `"P"("A") + 1/2 - 1/2"P"("A") = 2/3`
∴ `1/2"P"("A") = 1/6`
∴ P(A) = `1/3`
From (2),
P(B) + P(C) – P(B) P(C) = `3/4` ...[∵ B, C are independent]
∴ `"P"("B") + 1/2 - 1/2 "P"("B") = 3/4`
∴ `1/2"P"("B") = 1/4`
∴ P(B) = `1/2`
∴ P(A) = `1/3`, P(B) = P(C) = `1/2`
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