25. Laplace s rule of succession says that if there have been n Bernoulli trials and they have all been successes, then the probability of a success on the next trial is (n  1)/(n  2). For the derivation Laplace used a beta prior with a  1 and b  1 for binomial data, as in Example 14.7.

a. Show that, if a1 and b1 and there are n successes in n trials, then the posterior mean of p is

(n  1)/(n  2).

b. Explain

(a) in terms of total successes and failures;

that is, explain the result in terms of two prior trials plus n later trials.

c. Laplace applied his rule of succession to compute the probability that the sun will rise tomorrow using 5000 years, or n  1,826,214 days of history in which the sun rose every day. Is Laplace s method equivalent to including two prior days when the sun rose once and failed to rise once? Criticize the answer in terms of total successes and failures.