Try the following problem:
"Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what's behind the doors, opens another door [randomly from a door that does not have a prize behind it], say #3, which has a goat. He says to you, 'Do you want to pick door #2?' Is it to your advantage to switch your choice of doors? " (Problem as formulated by Craig F Whitaker)
The answer of most people is that it doesn't matter, since the two remaining doors are equally likely to contain the prize. However, this answer is wrong! The first goal of this project will to understand how to analyze this problem with mathematical clarity. This will lead to the basic concepts in the mathematics of chance, i.e. Probability theory.
It is also possible to get at the answer without using more than arithmetic. Play this game hundreds of times, and test two strategies: strategy A is to never switch the door, and strategy B is to switch always. See which strategy wins. Even better is to write a computer simulation of this game. Our second goal will be to write such simulations.
The mathematics involved here will lead to a discussion of Bayes' theorem, that has to do with probabilities of events relative to evidence. Our third goal will be to understand this theorem and use it to build a simple sound pattern or text pattern recognition system.
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