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Question
Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls? Given that
- the youngest is a girl.
- at least one is a girl.
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Solution
Let the first and second children be the girls G1, G2 and the boys be B1, B2
∴ S = {(G1, G2), (G1, B2), (G2, B1), (B1, B2)}
Let A = both children are girls = {G1, G2}
B = Youngest child is a girl = {(G1, G2), (B1, G2)}
C = At least one child is a girl = {(G1, B2), (G1, G2), (B1, G2)}
A ∩ B = {G1, G2},
A ∩ C = {G1, G2}
P(A ∩ B) = `1/4`, P(A ∩ C) = `1/4`
P(B) = `2/4`, P(C) = `3/4`
- P(A|B) = `(P(A ∩ B))/(P(B)) = 1/4 ÷ 2/4 = 1/2`
- P(A|C) = `(P(A ∩ C))/(P(C)) = 1/4 ÷ 3/4 = 1/3`
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