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Given a state of light, how do its properties change when only some of theconstituent photons are observed and the rest are neglected (traced out)? Bydeveloping formulae for mode-agnostic removal of photons from a beam, we showhow the expectation value of any operator changes when only $q$ photons areinspected from a beam, ignoring the rest. We use this to reexpress expectationvalues of operators in terms of the state obtained by randomly selecting $q$photons. Remarkably, this only equals the true expectation value for a uniquevalue of $q$: expressing the operator as a monomial in normally ordered form,$q$ must be equal to the number of photons annihilated by the operator. Auseful corollary is that the coefficients of any $q$-photon state chosen atrandom from an arbitrary state are exactly the $q$th order correlations of theoriginal state; one can inspect the intensity moments to learn what any randomphoton will be doing and, conversely, one need only look at the $n$-photonsubspace to discern what all of the $n$th order correlation functions are. Theastute reader will be pleased to find no surprises here, only mathematicaljustification for intuition. Our results hold for any completely symmetricstate of any type of particle with any combination of numbers of particles andcan be used wherever bosonic correlations are found.

Correlations for subsets of particles in symmetric states: what photons are doing within a beam of light when the rest are ignored