Probability of overlapping events with a I am trying to get the probability of overlapping time windows for a data readout. I have a process that occurs at rate R. The readout window of said process is split into three sections: Pre-condition (x seconds long), condition (x+1 seconds long, post-condition (x+2 seconds long). The condition windows cannot overlap, while the pre- and post-condition window can overlap. What is the probability of two windows of this process overlapping?
Time goes this way ->
Trigger 1
|----|-----|------|

Trigger 2
               |----|-----|------|

Trigger 3
                        |----|-----|------| 

Thanks a lot. 
 A: If your pre-condition interval is a fixed constant x, the process, once triggered, lasts x+1, and then there is a post-condition period of x+2 seconds long, then the probability of two processes overlapping can be calculated from the start times of two sequential instances of the process (i.e., $T_1, T_2$) using conditional probability:
$P_{(T_2\space overlaps\space T_1)|T_1}= P(T_2-(T_1+x+1)\leq(2x+2))$
We can define this probability as $P_o$. If the underlyling rate process is constant, then we have what is called a "renewal process", where after each event occur, the entire problem "resets", so we only need to know $P_o$ to answer your question, as overlaps are pairwise and events are sequential.
Since there is a fixed "dead zone" of length x+1 after each process starts, where another process cannot occur, I will interpret your average rate of occurrance, "R", as the rate per unit of "down-time", where a process is not currently underway. In this case, one model that is possible is the Poisson model with rate parameter R. The time between occurrances of this process will be exponentially distributed with the average time between the end of one process and the start of the next being $\frac{1}{R}$. Lets define this inter-process time as X, with distribution $P(X\leq t)=1-e^{-tR}$ 
Putting all this together, we get: $P_o = P(X\leq 2x+2)=1-e^{-(2x+2)R}$. This is the probability that a specific pair of events will overlap. If you have N process events, then you have N-1 inter-event intervals, and you can model the total number of overlapping windows as a Binomial random variable, which is a model of the number of Heads is a series of coin flips. In this case, Heads = Windows Overlap, Tails=Windows don't overlap. From the earlier calculation, we can see that $P(Heads)=P_o$. 
So, to get to your original question, the probability that exactly two windows overlap in N occurrances of the process is given by the binomial distribution, with $O=$ number of overlapping process windows:
$P(O=2|N-1)=Binomial(2,N-1,P_o)=$$N\choose2$$ (P_o)^2(1-P_o)^{N-3}$ the general forumula for $O\leq N$ is:
$P(O=x|N-1)=Binomial(x,N-1,P_o)=$$N\choose x$$ (P_o)^x(1-P_o)^{N-1-x}$
