I am working with certain recurring sequences in genetics and try to calculate certain probabilities:

Let for instance

$\displaystyle $$\langle g_i\rangle :=\{1,1,1,6,1,1,1,6,...,1,1,1,6\}$$ $ and

$\displaystyle $$\langle h_i\rangle :=\{1,1,1,1,6,1,1,1,1,6,...,1,1,1,1,6\}$$$

be to recurrent chains of identical length $\displaystyle $i_{\max}=20$$

How can one calculate the probability that at a certain selected index $\displaystyle $i_*$$ we would have on both chains a $\displaystyle $1$$, i.o.w.:

$\displaystyle $$\langle g_{i_*}\rangle =\langle h_{i_*}\rangle =1$$$

to explain further the expansion of the problem...

let $\displaystyle $\langle a_{i,m} \rangle$$ be $\displaystyle $m$$ recurring sequences (in above example $\displaystyle $m=\{1,2\}$$) each with different frequencies of recurrence $\displaystyle $f_m$$ (above example $\displaystyle $f_m=\{4,5\}$$) and the identical entrained length of $\displaystyle $L=\prod_{m}f_m$$.

How can I calculate the probability that (only) the 1s (the digits 1) of all $\displaystyle $m$$ chains would be compatible?