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  4. 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 :

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Q. 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 :

JEE MainJEE Main 2019Probability - Part 2

Solution:

$P(B) = P(G) = 1/2$
Required Proballity = $\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)} $
$ = \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} $