[M33] Consider a Bernoulli process with unknown probability p of a successful outcome. Assume that the prior probability of the bias p is uniformly distributed over the real interval (0, 1). Prove that after m successful outcomes in n independent trials, the expectation of a successful outcome in the (n + 1)th trial is given by (m + 1)/(n + 2).

Comments. This reduces to binary Bernoulli processes (p, 1 − p) with probability p of ‘success’ and probability 1 − p of ‘failure,’ that is, independent flips of a coin with unknown bias p. This is P.S. Laplace’s celebrated law of succession. Hint: The prior probability density P(X = p) is uniform with  b p=a P(X = p) = b − a (0 ≤ a ≤ b ≤ 1). The term Pr(Y =

m|n, p) = n m

pm(1−p)n−m is the probability of the event of m successes in n trials with probability p of success. The probability of obtaining m successes in n trials at all is Pr(Y = m|n) =  1 p=0 n m

pm(1 − p)n−mdp.

The requested expectation is the p-expectation of the posterior in Bayes’s rule, that is,  1 p=0 p P(Y |n, p)dp/P(Y = m|n). The integrals are beta functions; decompose these into gamma functions and use the relation of the latter to factorials. Source: [P.S. Laplace, A Philosophical Essay on Probabilities, Dover, 1952]. (Originally published in 1819. Translated from the 6th French edition.)