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Synthesis models predict the integrated properties of stellar populations.Several problems exist in this field, mostly related to the fact thatintegrated properties are distributed. To date, this aspect has been eitherignored (as in standard synthesis models, which are inherently deterministic)or interpreted phenomenologically (as in Monte Carlo simulations, whichdescribe distributed properties rather than explain them). We approachpopulation synthesis as a problem in probability theory, in which stellarluminosities are random variables extracted from the stellar luminositydistribution function (sLDF). We derive the population LDF (pLDF) for clustersof any size from the sLDF, obtaining the scale relations that link the sLDF tothe pLDF. We recover the predictions of standard synthesis models, which areshown to compute the mean of the sLDF. We provide diagnostic diagrams and asimplified recipe for testing the statistical richness of observed clusters,thereby assessing whether standard synthesis models can be safely used or astatistical treatment is mandatory. We also recover the predictions of MonteCarlo simulations, with the additional bonus of being able to interpret them inmathematical and physical terms. We give examples of problems that can beaddressed through our probabilistic formalism. Though still under development,ours is a powerful approach to population synthesis. In an era of resolvedobservations and pipelined analyses of large surveys, this paper is offered asa signpost in the field of stellar populations.Comment: Accepted by A&A. Substantially modified with respect to the 1st  draft. 26 pages, 14 fig

Confidence limits of evolutionary synthesis models