Derivation of time probability distributions for single molecule kinetics If we have a protein $\ce{A}$ that can bind to protein $\ce{B}$:

$$\ce{A + B &lt;=&gt;[$k_\mathrm{fwd,1}$][$k_\mathrm{rev,1}$] AB}$$

and the bound complex $\ce{AB}$ can be converted into $\ce{C}$:

$$\ce{AB &lt;=&gt;[$k_\mathrm{fwd,2}$][$k_\mathrm{rev,2}$] C}$$

Assume we have a volume of 1 and some initial amount of $\ce{A}$ and $\ce{B}$ (which are $A_0,$ $B_0$) and that $C_0 = 0.$ We then measure the amount of time elapsed before the first molecule of $C$ is formed. The experiment is repeated many times, and the frequency of occurence of first $C$ during each time interval 0 to $t_1$, $t_1$ to $t_2$, $t_2$ to $t_3$, etc. is plotted, so that the result is a probability distribution of the time elapsed before the first molecule of $C$ is formed. 

I would like to know what is the mathematical function that describes this distribution and how is it derived?

UPDATE: this is not a homework question and the point is not a specific answer. I would like references to literature/textbooks that derives these distributions analytically for simple examples like the above would be helpful. Have not seen a setup where distribution of duration is derived. Commenters say it's standard. Please give references.
To see how this calculation is done, let us consider the simpler case of just the second step shown above, ie $\ce{AB -&gt;[k_{f2}] C}$. Since we are interested in the situation where no $C$ has yet formed, the reverse reaction can be ignored. I will assume that this reaction is first order in $AB$, so $k_{f2}$ has units of $s^{-1}$. 

In this case, the rate constant $k_{f2}$ represents the average frequency of the reaction. 

Now let's consider that we establish an infinite number of evenly spaced time intervals of length $\Delta t$. When we consider the probability distribution, we need to consider two factors:

1) The probability of reaction occurring in an interval $t_n + \Delta t$ and

2) The probability that the reaction has not yet occurred at $t_n$

The first value is simply the product of the rate constant $k_{f2}$ and the length of the time interval: $k_{f2}\Delta t$.

The second probability is slightly more complicated, since it is the chance of the reaction $not$ occurring in each of the $n$ intervals preceding $t_n$. That probability is given by $1-(k_{f2}\Delta t)^n$. Conveniently, as our time interval $\Delta t$ goes to $0$, this probability becomes a continuous function $e^{-k_{f2}t}$.

To get our final probability, we simply multiply these two probabilities:

$[\text{total probability of reaction occurring in interval }t+dt] = [\text{probability of occurring in interval }t+dt] \times [\text{probability that reaction has not occurred by time }t]=k_{f2}dt\times e^{-k_{f2}t}$

The probability as a continuous function of $t$ therefore can be written as 

$p_{\ce{AB-&gt;C}}(t)=k_{f2}e^{-k_{f2}t}$.

As expected, integration from $t=0$ to $t=\infty$ gives an overall probability of $1$.

Reference: Adapted from the lecture notes published by Dmitrii Makarov here: http://makarov.cm.utexas.edu/resources/Lecture-notes,-tutorials-etc./single-vs-bulk.pdf

For the two-step process, if $\ce{A+B &lt;=&gt; AB}$ establishes equilibrium very fast relative to the $\ce{AB-&gt;C}$ reaction, then the above is a reasonable approximation (after adjustment of $k_{f2}$ to reflect the equilibrium concentration of $AB$). If the equilibrium is not established quickly, then you would use the same reasoning to determine the probability first of $AB$ having formed by time $t$ and then the probability of it reacting based on how long it has been in existence. 

I have not read Makarov's textbook on single molecule kinetics, but I assume it covers these kinetics in much greater detail and might be a good resource:

Single Molecule Science: Physical Principles and Models
1st Edition by Dmitrii E. Makarov; CRC Press 2015; ISBN 9781466559516