Martin Nymark: Från Bertrand till Gardner med Bayes sats

Abstract.

 In this thesis we examine how probability changes when we receive more information. With the help of Bayes’ theorem, we analyze several problems with discrete sample spaces that originated from the problem of Joseph Bertrand’s box paradox from the end of the 19th century. We discuss a few different solutions, one of which is a criterion for the likelihoods derived by Bayes’ theorem. When fulfilled, this criterion entails that two events in a conditional probability become independent.

In the middle of the 20th century, a variant of Bertrand’s boxes was popularized by Martin Gardner, most commonly known as The Boy or Girl Paradox, a problem that was ambiguously formulated. Two different solutions to Gardner’s problem are discussed, solutions that depend on how the new information was obtained.

Two newer variants of Gardner’s problem are analyzed. Assuming that the new information is obtained in a certain way, the solutions to these lead to surprising probabilities. We find that the results follow from events that are independent becoming conditionally dependent given the new information.