 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, givingIn particularThe 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.