Question

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

• A. $\frac{1}{4}$
• B. $\frac{2}{3}$
• C. $\frac{3}{4}$
• D. $\frac{1}{3}$

Answer 1

The correct option is D $\frac{1}{3}$

Let $b$ stands for a boy and $g$ for girl. The sample space for the experiment is
$S=\left\{\right(b,b),(g,b),(b,g),(g,g\left)\right\}$
Let $E$ and $F$ denote the following events:
$E:$ 'both the children are boys'
$F:$ 'at least one of the child is a boy'
Then $E=\left\{\right(b,b\left)\right\}$ and $F=\left\{\right(b,b),(g,b),(b,g),\}$
Now $E\cap F=\left\{\right(b,b\left)\right\}$
Thus $P\left(F\right)=\frac{3}{4}$ and $P(E\cap F)=\frac{1}{4}$
$\therefore$ $P\left(E/F\right)=\frac{P(E\cap F)}{P\left(F\right)}=\frac{\frac{1}{4}}{\frac{3}{4}}=\frac{1}{3}$