I am working with certain recurring sequences in genetics and try to calculate certain probabilities:
Let for instance $$\langle g_i\rangle :=\{1,1,1,6,1,1,1,6,...,1,1,1,6\}$$ and $$\langle h_i\rangle :=\{1,1,1,1,6,1,1,1,1,6,...,1,1,1,1,6\}$$
be to recurrent chains of identical length $i_{\max}=20$
How can one calculate the probability that at a certain selected index $i_*$ we would have on both chains a $1$, i.o.w.:
$$\langle g_{i_*}\rangle =\langle h_{i_*}\rangle =1$$
... to explain further the expansion of the problem... let $\langle a_{i,m} \rangle$ be $m$ recurring sequences (in above example $m=\{1,2\}$) each with different frequencies of recurrence $f_m$ (above example $f_m=\{4,5\}$) and the identical entrailed length of $L=\prod_{m}f_m$. How can I calculate the probability that (only) the 1 of all $m$ chains would be compatible?
Assuming you are picking a single value i, the arrays are of twenty elements and the pattern is identical to the one you have here, the answer is fairly easy, every fourth and fifth element, if picked will not be a match, there are 20/4= 5 fourth elements, plus 20/5= 4 fifth elements, so 9 pairs that are incompatible. So the probability of a match is (20-9)/20= 55%