Binding-plus-opening first-passage time ($\mathbb{E}[T]=(a+b+c)/(ac)$)

Consider unbound U, bound closed C and open O in a continuous-time Markov chain, with rates $U\to C=a$, $C\to U=b$, $C\to O=c$, and O absorbing over the observation interval. The mean first-opening times satisfy $T_U=1/a+T_C$ and $T_C=1/(b+c)+b{T}_U/(b+c)$, hence $T_U=(a+b+c)/(ac)$. Repeated unsuccessful binding attempts make this different from $1/c$. A FRET trace that cannot distinguish U from C does not by itself identify the individual binding and opening rates.