Question

Mark the correct alternative in the following question:
Assume that in a family, each child is equally likely to be a boy or a girl. A family with three children is chosen at random. The probability that the eldest child is a girl given that the family has at least one girl is

Options

• \[ \frac{1}{2} \]

• \[\frac{1}{3}\]

• \[\frac{2}{3} \]

• \[\frac{4}{7}\]

Answer 1

\[\text{ We have } , \]
\[S = \left\{ BBB, BBG, BGB, BGG, GGG, GBG, GGB, GBB \right\}, \text{ where the first letter in each element represents the eldest child } \]
\[\text{ Let} : \]
\[\text{ A be the event of choosing a family with a girl as the eldest child and } \]
\[ \text{ B be the event of choosing a family with at least one girl child } \]
\[\text{ So } , A = \left\{ GGG, GBG, GGB, GBB \right\} \text{ and } B = \left\{ BBG, BGB, BGG, GGG, GBG, GGB, GBB \right\}\]
\[ \Rightarrow n\left( A \right) = 4, n\left( B \right) = 7 \text{ and } n\left( A \cap B \right) = n\left( A \right) = 4\]
\[\text{ Now} , \]
\[P\left( A|B \right) = \frac{n\left( A \cap B \right)}{n\left( B \right)} = \frac{4}{7}\]