(because there are no singularities in the range of the posterior, and there are a finite number of terms) gives: So the posterior expectation for is: An approximate analytical expression for large N is given by first making the approximation to the product term: and then replacing the summation in the numerator with an integral The same procedure is followed for the denominator, but the process is a bit more tricky, as the integral is harder to evaluate where ln is the natural logarithm plugging in these approximations into the expectation gives where the base 10 logarithm has been used in the final answer for ease of calculation. For instance if the population is of size 10k then probability of success on the next sample is given by: So for example, if the population be on the order of tens of billions, so that k = 10, and we observe n = 10 results without success, then the expected proportion in the population is approximately 0.43%. If the population is smaller, so that n = 10, k = 5 (tens of thousands), the expected proportion rises to approximately 0.86%, and so on. Similarly, if the number of observations is smaller, so that n = 5, k = 10, the proportion rise to approximately 0.86% again. This probability has no positive lower bound, and can be made arbitrarily small for larger and larger choices of N, or k. This means that the probability depends on the size of the population from which one is sampling. In passing to the limit of infinite N (for the simpler analytic properties) we are "throwing away" a piece of very important information. Note that this ignorance relationship only holds as long as only no successes are observed. It is correspondingly revised back to the observed frequency rule as soon as one success is observed. The corresponding results are found for the s=n case by switching labels, and then subtracting the probability from 1. Generalization to any number of possibilities This section gives a heuristic derivation to that given in Probability Theory: The Logic of Science. The rule of succession has many different intuitive interpretations, and depending on which intuition one uses, the generalisation may be different. Thus, the way to proceed from here is very carefully, and to re-derive the results from first principles, rather than to introduce an intuitively sensible generalisation. The full derivation