= Multiplicity bias from a detector trigger
{title2=$\Pr(N=n\mid\mathrm{trigger})\propto A(n)\Pr(N=n)$}
A trigger changes which particle populations are common in the recorded sample. Let $N$ be the number of particles in an event, and stipulate an independent <Bernoulli distribution> for each particle's ability to produce an accepted telescope signal, with success <probability> $\epsilon$. If any successful particle suffices, the event acceptance is
$$
A(n)=1-(1-\epsilon)^n.
$$
This is a deliberately simplified model. Its $\epsilon$ includes the joint geometrical acceptance and response of the whole telescope; it is not an assumed universal single-counter efficiency. Actual shower directions, spatial correlations and shared electronics can violate the independence assumption.
The <conditional probability> formula gives
$$
\Pr(N=n\mid\mathrm{trigger})=
\frac{A(n)\Pr(N=n)}{\sum_m A(m)\Pr(N=m)}.
$$
For chosen underlying fractions $\Pr(N=1)=0.90$, $\Pr(N=5)=0.10$ and $\epsilon=0.15$, the two acceptances are $0.15$ and $0.5563$. Five-particle events consequently form about $29.18\%$ of recorded events. \b[A population that is only $10\%$ of incoming events can occupy nearly $30\%$ of the photographs.] This is <selection bias>, not evidence that the source population changed.
\Image[/coincidence-window-and-shower-selection.png]
{title=A coincidence window trades timing acceptance against background, while a trigger enriches events with more particles}
{description=Original calculations with the explicitly chosen rates, Gaussian timing spread and independent-particle acceptance model in the text. No historical event counts or measured timing curves are reproduced.}
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