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

1. the youngest is a girl.
2. at least one is a girl.

Answer 1

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`

1. P(A|B) = `(P(A ∩ B))/(P(B)) = 1/4 ÷ 2/4 = 1/2`
2. P(A|C) = `(P(A ∩ C))/(P(C)) = 1/4 ÷ 3/4 = 1/3`