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Bayesian Treasure Hunt

Bluebeard is known to have hidden a large part of his treasure on one of two Pacific islands named Palau and Nauru. It is believed to be twice as likely that the island is Palau instead of Nauru. A set of instructions to the location of the treasure has been found but most of the name of the island is blotted out. Only a letter u can still be recognized. Does this change the probability that the island is Palau?

The hypothesis is that the treasure is on Palau and the evidence is that one random letter from the name of the island is u. The prior probabilities of the hypothesis are given as \(P(H)=2/3\) and \(P(\overline{H})=1/3\). The likelihood that a random letter is u, given that the island is Palau, is \(P(E|H)=1/5\). The likelihood that a random letter is u, given that the island is Nauru, is \(P(E|\overline{H})=2/5\). The posterior probability of the hypothesis, given the evidence is then:

\[ P(H|E)=\frac{P(E|H)P(H)}{P(E|H)P(H)+P(E|\overline{H})P(\overline{H})} \]

\[ P(H|E)=\frac{(1/5)(2/3)}{(1/5)(2/3)+(2/5)(1/3)}=\frac{1}{2} \]

The probability is now a toss up between the islands. Both of them are equally likely.

This is a simple example of the kind of problem you can solve using Bayes' theorem. It comes from a new book of probability problems that we will be publishing soon. We are looking for people who want to review the book. If you are interested let me know (stefan at exstrom dot com) by Monday, March 18 2013.

© 2010-2013 Stefan Hollos and Richard Hollos

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