Question

A couple has two children, the older of which is a boy. What is the probability that they have two boys?

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{3}{4}$
• D. None of these

Answer 1

The correct option is A $\frac{1}{2}$

To find the probability that both are boys, if we know that the older child is boy.

Let $A$ be the event that both are boys.

Let $B$ be the event that older one is a boy.

$\therefore A=\{BB,BB\},B=\left\{BB\right\}$

$\Rightarrow P(A\cap B)=\left\{BB\right\}$

$P$(both are boys given the older child is a boy) $P\left(A/B\right)=\frac{P(A\cap B)}{P\left(B\right)}$

$S=\{BB,BG,GB,GG\}$ where $B$ represents boy, $G$ represents girl.

$\therefore P\left(A\right)=\frac{1}{4},P\left(B\right)=\frac{2}{4}=\frac{1}{2},P(A\cap B)=\frac{1}{4}$

$\Rightarrow P\left(B|A\right)=\frac{P(A\cap B)}{P\left(B\right)}=\frac{1/4}{2/4}=\frac{1}{2}$

Thus the probability that they have two boys is $\frac{1}{2}$.

(P(A∩Given S = {MM, MF, FM, FF}, we can see that: P (A)