\begin{aligned}
 p_{i} =\ & \frac{a+1}{a+b+1}\cdot p_{i-1} + \frac{a}{a+b+1}\cdot q_{i-1} \\
 q_{i} =\ &\frac{b}{a+b+1}\cdot p_{i-1} + \frac{b+1}{a+b+1}\cdot q_{i-1}
\end{aligned}