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Classic Problems of Probability by Prakash Gorroochurn

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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 p1 and p2 is

img

Solution. Let Xn 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,

img

To find img, let us first obtain the conditional density img by using Bayes' Theorem1:

img

Now, we have2

img

Therefore,

img

Hence,

img

as required.

14.1 Discussion

This is essentially ...

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