Question

Assume that each child born 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 first and second girls be denoted by G1 and G2 and boys B1 and B2.

∴S = {(G1, G2), (G1, B2), (G2,B1), (B1, B2)}

Let,
A = Both the children are girls = (G1, G2)
B = Youngest child is girl = {(G1, G2), (B1, G2)}
C = at least one is a girl = {(G1, B2),(G1, G2),(B1, G2)}

A∩B = (G1, G2)

⇒P(A∩B) = $\frac 14$

A∩C = (G1, G2)
⇒P(A∩C) = $\frac 14$

P(B)=$\frac 24$ and P(C)=$\frac 34$

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$(i)$ $P(A|B) = \frac {P(A∩B)}{P(B)}$

$P(A|B)=\frac {1/4}{2/4}$

$P(A|B)=\frac 12$

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$(ii)$ $P(A|C)=\frac {P(A∩C)}{P(C)}$

$P(A|C)=\frac {1/4}{3/4}$

$P(A|C)=\frac 13$

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