 Finite-population model of raven confirmation

Fix $r>1$ ravens and $M>1$ nonblack objects in a finite population. Under $H_0$, all ravens are black; under $H_1$, exactly one is nonblack. The total numbers $r,M$ are held fixed under both hypotheses, so the classification of one nonraven also changes between hypotheses. This is a deliberately specified model for Bayesian confirmation, not a universal rule about how bird observations supply evidence.

A uniform draw among nonblack objects returns a nonraven with probabilities $1$ under $H_0$ and $1-1/M$ under $H_1$. A uniform draw among ravens returns a black object with probabilities $1$ and $1-1/r$. The respective likelihood ratios for $H_0$ over $H_1$ are therefore $(1-1/M{)}^{-1}$ and $(1-1/r{)}^{-1}$. When $M>r$, the former is closer to one, so supplies less support under these assumptions.

With equal prior probabilities, Bayes' theorem gives posterior probabilities $1/(2-1/M)$ and $1/(2-1/r)$. More generally, with prior probability $p$ for $H_0$, replace the two expressions by $p/[p+(1-p)(1-1/M)]$ and $p/[p+(1-p)(1-1/r)]$. Previously known information or a different sampling protocol requires different likelihoods; selecting an object already known to be a nonraven need not supply new evidence.