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
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,
To find , let us first obtain the conditional density by using Bayes' Theorem1:
Now, we have2
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