I'm reading this paper, which describes (among other things) the "triggering" system for a peice instrumentation used in the detection of subatomic particles in an astroparticle physics experiment.
The experiment is subject to thermal noise, which can be modeled as a Gaussian with $\mu = 0$ and $\sigma = V_{rms}$. So, given some sampling rate, a waveform can be simulated by making a random pick from the Gaussian for each sample.
The instrument in question has $n$ channels, and a "triggering" system which is designed to record every time $k$ channels have an excursion over some threshold within a duration $\tau$. So, for example, a trigger may be defined as $3$ channels having an excursion over $5$ within $10$ nanosecond of the first such excursion.
The paper then describes a method to determine this trigger rate, which really confounds me:
To determine the expected accidental rate, RL1, of $k$-fold coincidences among the $n$ = 8 channels, consider a trial event, defined by a hit in any one of the $n$ channels, which then triggers a logic transition out of the discriminator to the logic TRUE state for a duration τ. Then consider the probability during this trial that $k > −1$ or more (k = 3 for [this experiment]) additional sub-band discriminator logic signals arrive while the first is still in TRUE state, corresponding to a hit above threshold for that channel. The rate of TRUE states per channel is $r$. We do not for now assume $rτ << 1$. The probability to observe exactly $k − 1$ out of $n − 1$ additional channels in the TRUE state after one channel has changed its state is given by the binomial (e.g., the $k$ out of n ‘coin toss’) probability:
$P(k-1 : n-1) = \frac{(n-1)!}{(k-1)!(n-k)!}p^{k-1}(1-p)^{n-k}$
The single channel ‘coin-toss’ probability p is just given by the fractional occupancy of the TRUE state per second per channel: $p = > rτ$. The probability per trial to observe greater than $k − 1$ out of $n − 1$ channels is then just the cumulative probability density of the binomial distribution times the observation interval:
$P(\geq k-1 : n-1) = \sum_{j=k-1}^{n-1} \frac{(n-1)!}{(j)!(n-1-j)!}(r\tau)^j[1-r\tau]^{n-1}dt$
For $rτ << 1$ as it often is in practice, this simplifies to:
$P(\geq k-1 : n-1) \approx \frac{(n-1)!}{(k-1)!(n-k)!}(r\tau)^{k-1}$
...since only the leading term in the sum contributes significantly and the term $1−rτ ≃ 1$. The rate is then determined by multiplying the single-trial probability by the number of ensemble trials per second, which is just equal to the total number of channels times the singles rate per channel. The singles rate per channel is given simply by $r$, and the total singles rate across all channels is $nr$. Thus the total rate in the limit of $rτ << 1$, is:
$\text{rate (RL1)} = nrP(\geq k-1:n-1) \approx n\frac{(n-1)!}{(k-1)!(n-k)!}r^k\tau^{k-1} = \frac{(n)!}{(k)!(n-k)!}kr^k\tau^{k-1}$
I can follow most of the logic (the first equation is (as stated) just the binomial probability formula, the second follows trivially from the first), however some of this seems "pulled out of thin air" to me (namely, the line involving the summation, and everything after).