So the context is that we have an Urn with $N$ balls, $M$ red balls and $N-M$ white balls and we are sampling without replacement. All this information is denoted by proposition $B$ and $R_i$ is the proposition that we have a red ball on $ith$ draw. Then Jaynes writes
But Jaynes does not provide any explicit justification for $P(R_1|B) = E(M/N)$ or maybe I am missing something implicit. Anyone familiar with this help me out.
If $M,N$ are random variables that have a joint distribution then by the law of total probability:$$P(R_1|B)=\sum_{n,m}P(R_1|B,M=m, N=n)P(M=m,N=n)=$$$$\sum_{n,m}\frac{m}{n}P(M=m,N=n)=\mathbb E\left(\frac{M}{N}\right)$$