Problem 14

Bayes, Laplace, and Philosophies of Probability (1764, 1774)

Problem. Assume that the probability p of an event A is equally likely to be any number between 0 and 1. Show that, conditional on A having previously occurred in a out of n independent and identical trials, the probability that p lies between p_{1} and p_{2} is

Solution. Let X_{n} be the number of times that the event A occurs in n trials. Since p is equally likely to be between 0 and 1, we let p have a probability density function that is uniform on [0,1], that is,

To find , let us first obtain the conditional density by using Bayes' Theorem^{1}:

Now, we have^{2}

Therefore,

Hence,

as required.

# 14.1 Discussion

This is essentially ...