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Question
A family has two children. Find the probability that both the children are girls, given that atleast one of them is a girl.
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Solution
A family has two children.
∴ Sample space S = {BB, BG, GB, GG}
∴ n(S) = 4
Let event A: At least one of the children is a girl.
∴ A = {GG, GB, BG}
∴ n(A) = 3
∴ P(A) = `("n"("A"))/("n"("S")) = 3/4`
Let event B: Both children are girls.
∴ B = {GG}
∴ n(B) = 1
∴ P(B) = `("n"("B"))/("n"("S")) = 1/4`
Also, A ∩ B = B
∴ P(A ∩ B) = P(B) = `1/4`
∴ Required probability = `"P"("B"/"A")`
= `("P"("B" ∩ "A"))/("P"("A"))`
= `(1/4)/(3/4)`
= `1/3`.
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