# 11.12 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              (ii) at least one is a girl?

S seema garhwal

Assume that each born child is equally likely to be a boy or a girl.

Let  first and second girl are denoted by $G1\, \, \, and \, \, \,G2$  respectively also  first and second boy are denoted by $B1\, \, \, and \, \, \,B2$

If a family has two children, then total outcomes $=2^{2}=4$$=\left \{ (B1B2),(G1G2),(G1B2),(G2B1)\right \}$

Let A= both are girls $=\left \{(G1G2)\right \}$

and  C= at least one is a girl =$=\left \{(G1G2),(B1G2),(G1B2)\right \}$

$A\cap B=\left \{(G1G2)\right \}$

$P(A\cap B)=\frac{1}{4}$                            $P( C)=\frac{3}{4}$

$P(A| C)=\frac{P(A\cap C)}{P(C)}$

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

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

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