Official solution
By Bayes' theorem:
\[P(\overline S\mid A) = \frac{P(A\mid \overline S)P(\overline S)}{P(A\mid \overline S)P(\overline S)+P(A\mid S)P(S)} = \frac{\frac{{}^{13}C_2}{{}^{51}C_2}\cdot\frac34}{\frac{{}^{13}C_2}{{}^{51}C_2}\cdot\frac34+\frac{{}^{12}C_2}{{}^{51}C_2}\cdot\frac14} = \frac{39}{50}\]