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Question
A family has two children. What is the probability that both the children are girls given that at least one of them is a girl?
Solution
Let B denote a boy and G denote a girl.
Then the sample, S = {BG, GB, BB, GG}.
∴ n(S) = 4
Let E be the event that both children are girls.
Let F be the event that at least one of them is a girl.
Then E = {GG}, n(E) = 1
F = {BG, GB, GG}, n(F) = 3
P(F) = `("n"("F"))/("n"("S")) = 3/4`
E ∩ F = {GG}, n(E ∩ F) = 1
P(E ∩ F) = `("n"("E" ∩ "F"))/("n"("S")) = 1/4`
Required Probability `"P"("F"/"E") = ("P" ("E" ∩ "F"))/("P"("F")) = (1/4)/(3/4) = 1/3`
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