The Monty Hall Problem: Newcastle, 2000, Australia

Prof. Graham Priest gave a talk at the Newcastle University (http://www.newcastle.edu.au/) in 2000 which made us think that the Monty Hall Problem (http://www.nytimes.com/2008/04/08/science/08monty.html) was a problem worth studying because it defied the laws of Mathematics.

The problem appears to be connected to a TV show where a car would be hidden behind a door. The other doors would have something else behind them. The person participating in the game would choose one of the three doors and, if they guessed right, like if they chose the door that had the car behind it, then they would get the car. The presenter would then open one of the two doors that remained (not chosen) after the person made the first choice. That door would never conceal the car. 


It is then said that they have studied the problem from the TV show and if the person changed their choice at that stage, like after the revelation of the contents of one of the two remaining doors, then the person would have more chances of getting the car because studying the history of the show they reached that conclusion, like, statistically, that would be the case.

A fellow from Linkedin listed all possible first choices and then drew a map of possibilities, what we call tree, to prove that truth. 

The problem is that his map is incomplete.

Please notice that we have three choices available the first time the question is asked and therefore one chance in three of finding the car (about 33% or 1/3). 

One of the doors is open and no car appears. 


We are now left with two doors. One of them has the car. 



Has the probability changed? Yes, sure! Now we have 50% or 1/2 of chance of getting the car, like it is more likely that we get it, since one door has been eliminated from the game and did not conceal it. 


What happens if we change our choice? We then have 50% of chance in the same way, is it not?

Notice that the first choice will never have any effect on the second because the presenter always opens a door that does not conceal the car. 

Notice that the door is not closed again and mixed, as we do with the cards game, and we have not eliminated the possibility of getting a non-car by opening that door. 


We then have about 33% of chance with the first choice. If we change the choice after the door is open, then we have 50% of chance of getting it and if we stick to the same door, we also have 50% of chance of getting it. 

Why is it that the history of choices of the show seem to conflict with this conclusion then? As in Brazil, with Silvio Santos, more than likely (same sort of game), we can only assume that people would change doors more rarely and it was then the choice of the production to swap the position of the car so that the person would lose, like that is just a TV show… . 

If we were to write a tree for this problem, it would look like this:

So, this is our complete tree of choices.

Basically, the first balls are the choice of the contestant on the first time and after that choice they will be presented with three doors again, but one will be eliminated, which is the one that appears with the cross.

With that, we have the following couples of results as our possibilities set:

{(D1,D1), (D1,D2),(D1,D1),(D1,D3),(D2,D2),(D2,D3),(D2,D1),(D2,D2),(D3,D1), (D3,D3), (D3,D2), (D3,D3)}

The cardinality of this set is 12. 

Notice that in six of the possibilities there is equality between the first and the second choice, what then gives 50% of the results as possible. If we swap, that is, if the second choice or the second member of the couple changes, then we have the other 50%.

We have then proven, also through tree of possibilities, that the chances are 50% for those who stick to the same door and 50% for those who change on the second opportunity.

We think that this settles the possible problem with the Mathematics involved.

Were the game honest, we probably should notice some similarity of wins between the people who swap and those who stick. Notice however that luck is luck and some numbers get out of the globe/bag more times than others with draws controlled by auditors in the popular games we have, say the game Lotto in Brazil. 


If we could draw a rule that were deterministic, this would be purely mathematical, not also statistical (probability falls to the side of Statistics). Whatever is statistical is not supposed to be deterministic by nature, so that the results are not predictable in terms of real life. The best we get is a percentage in terms of chances of getting it right.

Also, if we could predict results, then we would always win, what would obviously make profit impossible to the side of the organization running the game.

This problem perhaps exists to illustrate the differences between what is statistical and what is mathematical in the same way that The Sorites Paradox exists to illustrate the difference between purely human language and machine or mathematical language (Mathematics is based on a logical system that is very similar to the Classical Logic, which was the first logical system used by machines. The first circuits we had were all based on Classical Logic, and that is why they had only two states: Open or closed. Basically, to have our air-conditioners, the modern ones, we had to use Fuzzy Logic, which is a logical system that accepts more nuances, not only two states: gradations were hot and cold only before they came up with the dial, where we could choose something in between).

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