= Solution
Normalize the total channel density to one. For <independent permissive gates in an ion channel>, the <probability> that the $M$ gate is open is $m$, and the <probability> that the $H$ gate is open is $h$. The joint <probability> is the product of the appropriate open or closed <probabilities>, giving
$$
\boxed{S_{ij}=m^i(1-m)^{1-i}h^j(1-h)^{1-j}.}
$$
In particular
$$
S_{00}=(1-m)(1-h),\quad S_{10}=m(1-h),\quad
S_{01}=(1-m)h,\quad S_{11}=mh.
$$
The original PDF really prints $j$, rather than $1-j$, on the final factor. That expression fails <normalization of a probability distribution> and does not give the closed-$H$ <probability> at $j=0$; the displayed correction is necessary. If $S_{ij}$ are literal densities with conserved total density $N$, multiply each expression by $N$.
The product additionally assumes initially <independent events> and independent gate transitions. The transition scheme alone permits initially correlated gates and does not force this factorization at every time. The all-closed initial preparation used below satisfies the required product assumption.
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