Question

A family has two children. What is the probability that both the children are boys given that at least one of them is a boy?

Answer 1

Given: A family has two children.
Let the girls be denoted by ${}^{\prime }{g}^{\prime }\&$ boys be denoted by ${}^{\prime }{b}^{\prime }$.
Therefore,sample space of the experiment is
$s=\left\{\right(g,g),(g,b),(b,g),(b,b\left)\right\}$
Let $E:$ both the children are boys $\&$
$F:$ at least one of the child is a boy.
So,$E=\left\{\right(b,b\left)\right\}$
$P\left(E\right)=\frac{1}{4}$ ....$\left(1\right)$
Now,
$F:$ At least one of the children is a boy
$F=\left\{\right(g,b),(b,g),(b,b\left)\right\}$
$P\left(F\right)=\frac{3}{4}$ ....$\left(2\right)$
Also,
$E\cap F=\left\{\right(b,b\left)\right\}$
$P(E\cap F)=\frac{1}{4}$ .....$\left(3\right)$
$P\left(\frac{E}{F}\right)=\frac{P(E\cap F)}{P\left(F\right)}$ ....$\left(4\right)$
Substituting values of eq.$\left(2\right)$ $\&$ eq.$\left(3\right)$ in eq.$\left(4\right)$
$P\left(\frac{E}{F}\right)=\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right)$
$\therefore P\left(\frac{E}{F}\right)=\frac{1}{3}$
therefore, probability that both the children are boys given that at least one of them is a boy is $=\frac{1}{3}$