Question

A couple has two children, Find the probability that both children are males, if it is known that at least one of the children is male.

Answer 1

Sample space: $S=\{MM,MF,FM,FF\}$, where $M=$ male and $F=$ female.

To find $P\left(bothchildrenaremalesifitisknownthatatleastoneofthechildrenismale\right)$

$A:$ Event that both children are male, and

$B:$ event that at least one of them is a male

$A:\left\{MM\right\}$ and $B:\{MF,FM,MM\}$

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

Probability that both are males, if we know one is a female $=P\left(A|B\right)=\frac{P(A\cap B)}{P\left(B\right)}$

Given:$S=\{MM,MF,FM,FF\}$, we can see that

$P\left(A\right)=\frac{1}{4}$,$P\left(B\right)=\frac{3}{4}$,$P(A\cap B)=\frac{1}{4}$

$\therefore P\left(A|B\right)=\frac{P(A\cap B)}{P\left(B\right)}=\frac{\frac{1}{4}}{\frac{3}{4}}=\frac{1}{3}$