Question

A couple has two children. Find the probability that both the children are (i) males, if it is known that at least one of the children is male. (ii) females, if it is known that the elder child is a female.

Answer 1

Consider the given events.
A = Both the children are female.
B = The elder child is a female.
C = At least one child is a male.
D = Both children are male.

\[\text{ Clearly } , \]
\[S = \left\{ M_1 M_2 , M_1 F_2 , F_1 M_2 , F_1 F_2 \right\}\]
\[A = \left\{ F_1 F_2 \right\}\]
\[B = \left\{ F_1 M_2 , F_1 F_2 \right\}\]
\[C = \left\{ M_1 F_2 , F_1 M_2 , M_1 M_2 \right\} \]
\[D = \left\{ M_1 M_2 \right\}\]                                    

[Here, first child is elder and second is younger]

\[D \cap C = \left\{ M_1 M_2 \right\} \text{ and } A \cap B = \left\{ F_1 F_2 \right\}\]

\[\left( i \right) \text{ Required probability }  = P\left( D/C \right) = \frac{n\left( D \cap C \right)}{n\left( C \right)} = \frac{1}{3}\]

\[\left( ii \right) \text{ Required probability }  = P\left( A/B \right) = \frac{n\left( A \cap B \right)}{n\left( B \right)} = \frac{1}{2}\]