Closed form of a matrix product

Question

Is there any closed form or a bound for a matrix product of this kind $$ P=\prod_{i=1}^n \begin{pmatrix} 1-a & a \\ b_i & 1-b_i \end{pmatrix}, \quad a,b_i \in [0,1] $$ for an arbitrary sequence $\{b\}_i$? Essentially, I am trying to estimate a distribution of non-homogeneous Markov chain at time $n$, specifically, I would like to find the upper left corner of $P$: $$\begin{pmatrix} 1 & 0 \end{pmatrix} P \begin{pmatrix} 1 \\ 0 \end{pmatrix}.$$