Multiplicity bias from a detector trigger ($\mathrm{Pr}(N=n\mid \mathrm{trigger})\propto A(n)\mathrm{Pr}(N=n)$)

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 $ϵ$. If any successful particle suffices, the event acceptance isThis is a deliberately simplified model. Its $ϵ$ 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 givesFor chosen underlying fractions $\mathrm{Pr}(N=1)=0.90$, $\mathrm{Pr}(N=5)=0.10$ and $ϵ=0.15$, the two acceptances are $0.15$ and $0.5563$. Five-particle events consequently form about $29.18\%$ of recorded events. 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.