If the $5^{\mathrm{t}\mathrm{h}}$ ball is a red ball $\Rightarrow$ either the first four ball are of other colour and $5^{\mathrm{t}\mathrm{h}}$ ball is red or the first four ball consist of $1$ red ball, $3$ other ball and $5^{\mathrm{t}\mathrm{h}}$ ball is red ball
Hence, the required probability $=\frac{{}_{}{}^{9}C{}_4^{}}{{}_{}{}^{11}C{}_4^{}}\times \frac{2}{7}+\frac{{}_{}{}^{9}C{}_3^{}\times {}_{}{}^{2}C{}_1^{}}{{}_{}{}^{11}C{}_4^{}}\times \frac{1}{7}$
$=\frac{9\times 8\times 7\times 6}{11\times 10\times 9\times 8}\times \frac{2}{7}+\frac{9\times 8\times 7\times 2\times 4}{11\times 10\times 9\times 8}\times \frac{1}{7}$
$=\frac{6}{55}+\frac{4}{55}=\frac{10}{55}=\frac{2}{11}$