Question

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

Answer 1

$E:$ Both the children are boys

$E=\left\{\right(b,b\left)\right\}$

$P\left(E\right)=\frac{1}{4}$

$F:$ At least one of the children is a boy

$F=\left\{\right(g,b),(b,g),(b,b\left)\right\}$

$P\left(F\right)=\frac{3}{4}$

Also,

$E\cap F=\left\{\right(b,b\left)\right\}$

$P(E\cap F)=\frac{1}{4}$

Now,

$P\left(E|F\right)=\frac{P(E\cap F)}{P\left(F\right)}$

$=\frac{\frac{1}{4}}{\frac{3}{4}}=\frac{1}{3}$