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For $x = {1, 2, 3, ... , k}$ and

given $0<p_x<1,x\in(0,1)\text{ and }y\in(0,1)$ and $p_x$ values for all $x$ are known

$$ P(a_x|b_x,p_x)= \begin{cases} 1-p_x, &\text{if } (a_x,b_x)=(0,0)\\ p_x, &\text{if } (a_x,b_x)=(0,1)\\ p_x, &\text{if } (a_x,b_x)=(1,0)\\ 1-p_x, &\text{if } (a_x,b_x)=(1,1)\\ \end{cases} $$

\begin{align*}P(a_{(1\ldots n)}\mid b_{(1\ldots n)}) &= P(a_{(1\ldots n-1)}\mid b_{(1\ldots n-1)} = m-1,p_{(1\ldots n-1)})\times P(a_n\mid b_n=1,p_n) \\ &+P(a_{(1\ldots n-1)}\mid b_{(1\ldots n-1)} = m,p_{(1\ldots n-1)})\times P(a_n\mid b_n=0,p_n) \end{align*}

Question: How do I get $P(a_{(1\ldots k)} \mid b_{(1\ldots k)}=m,p_{(1\ldots k)})$ given conditions and equations?

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