## probabilities in recurring sequneces

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?