Haemophilia is a recessive X-linked trait. Let \(X^H\) represent the normal allele and \(X^h\) represent the allele for haemophilia.
A normal couple has a haemophilic son, which means the son's genotype is \(X^hY\). He inherited the \(X^h\) allele from his mother. Since the mother is normal but has an affected son, her genotype must be heterozygous carrier, \(X^HX^h\).
The father is normal, so his genotype is \(X^HY\).
Let's consider the possible genotypes of their offspring:
\begin{itemize}
\item Daughters: \(X^HX^H\) (normal) or \(X^HX^h\) (carrier). There is a 50% chance for each. The question states half the daughters are carriers, which aligns with the mother being \(X^HX^h\).
\item Sons: \(X^HY\) (normal) or \(X^hY\) (haemophilic). There is a 50% chance for each. The question states half the sons are haemophilic, which also aligns with the mother being \(X^HX^h\).
\end{itemize}
The gene for haemophilia is located on the X-chromosome. The mother, being a carrier (\(X^HX^h\)), passes on the \(X^h\) allele to half of her sons, making them haemophilic, and passes on the \(X^h\) allele to half of her daughters, making them carriers (\(X^HX^h\)). The father's X-chromosome (\(X^H\)) contributes to normal sons and normal or carrier daughters. Therefore, the mother carries the recessive allele on one of her X-chromosomes.