# Monty Hall Problem with Maple 17

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Let’s say you wanted to solve once and for all the Monty Hall 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 No. 1,

and the host, who knows what’s behind the doors, opens another door, say No.

3, which has a goat. He then says to you, Do you want to pick door No. 2?” Is

it to your advantage to switch your choice?

This problem has been debated for years. If you look it up on the web, you can find various

explanations of why it is better to switch your choice. The proof by using probability is not

obvious, so we will use our method of random number simulations to explore the following

questions:

The fi rst part of your fi nal project is to provide evidence of why you should switch

your choice in the three door game by using random number simulations.

The second part of project is to perform a similar experiment using a total of four

doors. (Only one has a car. The rest have goats.) In this case, there are two rounds

of opening a door (showing a goat) and each time you have the opportunity to switch

your choice. By simulation, can you determine the “most winning” strategy? It can

be a combination of switching or not switching in those two steps.

Can you extend your result to games with more doors? Do you think there is a pattern

as you add more doors?

A complete report would have the following outline:

Three Door Problem

{ Describe the problem. Look up an explanation of the (theoretical) probability of

winning for each choice and describe the reasoning in your own words.

{ Set up an experiment (simulation) for this game. Provide a copy of the code in

the Appendix.

{ Approximate the number of wins per total games for the switching strategy. Make

sure to report the number of simulations you used and repeat the experiment to

verify the results. Does this match the probability you predicted above?

{ Approximate the number of wins per total games for the NOT switching strategy.

Does this match the probability you predicted above?

{ Summarize your results and how they support your winning strategy.

Four Door Problem

{ Describe the problem and the possible switching strategies in your own words.

{ Set up an experiment (simulation) for this game. Provide a copy of the code in

the Appendix.

{ Approximate the number of wins per total games for each switching strategy.

Make sure to report the number of simulations you used and repeat the experiment

to verify the results.

{ From your data, conclude which is the best strategy. Can you explain why this

is true?

{ Can you extend your winning strategy to more doors? Why does it make sense?

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