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Mathematics Question on Probability

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 :

A

111\frac{1}{11}

B

117\frac{1}{17}

C

110\frac{1}{10}

D

112\frac{1}{12}

Answer

111\frac{1}{11}

Explanation

Solution

P(B)=P(G)=1/2P(B) = P(G) = 1/2
Required Proballity = all 4 girls(all 4 girls)+(exactly 3 girls+1 boy)+(exactly 2 girls+2 boy)\frac{\text{all} \, 4 \, \text{girls}}{\left(\text{all} \, 4 \, \text{girls}\right)+\left(\text{exactly}\, 3 \,\text{girls}+1\, \text{boy}\right)+ \left(\text{exactly}\, 2 \,\text{girls}+2\, \text{boy}\right)}
=(12)4(12)4+4C3(12)4+4C2(12)4=111= \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}