Possible typo in Jaynes' Probability Theory 6.44

Question

I read about this, $$ p(R\mid r=0,NI_{1}) = {N\choose n+1}{N-R\choose n} \tag{6.44}$$ but the probability can not be greater than 1, this expression is an apparent error. $I_{1}$ means the state of knowledge about the urn is that 0 < R < N, while the general sum $S = \sum_{R=0}^N{{R\choose0}{N-R\choose n-r}}= {N+1\choose n+1} $ is given by $I_{0}$ which means we do not know the state of the knowledge about urn at all, so the $S'= S - {N\choose n}\delta(r,0) - {N\choose n}\delta(r,n)$, and the S' should be the denominator. Finally i get the result is $$ p(R\mid r=0,NI_1) = \frac{N-R\choose n}{N\choose n +1}$$

Answer 1

Yes, you are right. Immediately before formula (6.44) the author states that (6.44) will replace (6.17). In (6.17) we see that he uses the notation $$\dbinom{N+1}{n+1}^{-1}\cdot\text{other terms}$$ Based on the correctness of your calculations and the notation of the author, (6.44) was probably intended to be $$p(R\mid r=0,NI_1)=\dbinom{N}{n+1}^{-1}\dbinom{N-R}{n}, \qquad 1\le R\le N-1$$ So, the author missed this $-1$ in the exponent.