JEE MAIN - Mathematics (2019 - 10th April Morning Slot - No. 1)
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 :
$${1 \over {10}}$$
$${1 \over {17}}$$
$${1 \over {11}}$$
$${1 \over {12}}$$
Explanation
A = At least two girls
B = All girls
$$P\left( {{B \over A}} \right) = {{P\left( {B \cap A} \right)} \over {P\left( A \right)}}$$
$$ \Rightarrow {{P(B)} \over {P(A)}} = {{{{\left( {{1 \over 4}} \right)}^2}} \over {1 - {}^4{C_0}{{\left( {{1 \over 2}} \right)}^4} - {}^4{C_1}{{\left( {{1 \over 2}} \right)}^4}}}$$
$$ \Rightarrow {1 \over {16 - 1 - 4}} = {1 \over {11}}$$
B = All girls
$$P\left( {{B \over A}} \right) = {{P\left( {B \cap A} \right)} \over {P\left( A \right)}}$$
$$ \Rightarrow {{P(B)} \over {P(A)}} = {{{{\left( {{1 \over 4}} \right)}^2}} \over {1 - {}^4{C_0}{{\left( {{1 \over 2}} \right)}^4} - {}^4{C_1}{{\left( {{1 \over 2}} \right)}^4}}}$$
$$ \Rightarrow {1 \over {16 - 1 - 4}} = {1 \over {11}}$$
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