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How does new evidence change our estimates of probabilities? Carnap's formula revisited

Suppose that we are in a situation where we need to know the probability of some event, but we cannot use standard statistical methods because we do not have enough observations. Assume that we have p 0, which is an expert estimate of that probability. We then perform N experiments in which this event occurs M times. In case N is so small that we still cannot use standard statistical methods, what is now a reasonable estimate for the probability? We show that reasonable conditions lead to the formula $ which was originally proposed by R. Carnap in his analysis of induction. A natural generalization of this formula is also presented.