{\displaystyle {\begin{aligned}P(s,f\mid q=x)&={s+f \choose s}x^{s}(1-x)^{f},\\P(q=x)&={x^{\alpha -1}(1-x)^{\beta -1} \over \mathrm {B} (\alpha ,\beta )},\\P(q=x\mid s,f)&={\frac {P(s,f\mid x)P(x)}{\int P(s,f\mid y)P(y)dy}}\\&={{{s+f \choose s}x^{s+\alpha -1}(1-x)^{f+\beta -1}/\mathrm {B} (\alpha ,\beta )} \over \int _{y=0}^{1}\left({s+f \choose s}y^{s+\alpha -1}(1-y)^{f+\beta -1}/\mathrm {B} (\alpha ,\beta )\right)dy}\\&={x^{s+\alpha -1}(1-x)^{f+\beta -1} \over \mathrm {B} (s+\alpha ,f+\beta )},\end{aligned}}}