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

Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is :

1. $\frac{1}{11}$

2. $\frac{1}{17}$

3. $\frac{1}{10}$

4. $\frac{1}{12}$

Correct Option: 1

Solution:

$P(B)=P(G)=1 / 2$

Required Proballity =

$\frac{\text { all } 4 \text { girls }}{(\text { all } 4 \text { girls })+(\text { exactly } 3 \text { girls }+\text { lboy })+(\text { exactly } 2 \text { girls }+2 \text { boys })}$

$=\frac{\left(\frac{1}{2}\right)^{4}}{\left(\frac{1}{2}\right)^{4}+{ }^{4} C_{3}\left(\frac{1}{2}\right)^{4}+{ }^{4} C_{2}\left(\frac{1}{2}\right)^{4}}=\frac{1}{11}$