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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