More Monty

Monty Hall rises again.  It has been some time since I have taken an interest in the Mont Hall problem.  I have no intention of reiterating the original problem as I have done this in the past and anyone interested can look it up.  It has been a problem that has bothered me for a long time.  It nags at the back of my mind regularly.  I spent a day or so of my mental CPU cycles I had spare taking a fresh look.

But this time, I am not so interested in the maths.  I have the spreadsheet produced by a friend of mine (spreadsheet and maths genius) Dan and it shows quite clearly the reality.  So if the maths is right (and it is) what is that itch in my brain?  It's this.  What does it mean?  If the maths is right, what is the maths telling us?  Is the surprise and debate in the maths world valid or is this in fact maths slight of hand?  After all, maths produces a model of the world where the emphasis is on 'a' but maybe not the and its claim to truth is not always justified.  It can be right (mathematically) but not necessarily be reality.  Of course, this gets trickier when we deal with probability like in the Monty Hall problem as probability and what it represents is not always clear and the subject of much philosophical discussion.

The standard 3 door Monty Hall problem has a clear result.  If you select a door at random, then Monty opens a door without a car behind it (it matters not what is behind it) then if you switch your choice of door from the original the outcome is that you will win 66.666.% of the time vs 33.333.% if you stick.  Note that I chose my words careful.  The outcome 'IS'…. 

I ran some other stats on the model.  Not surprisingly if you randomly select a door after Monty has opened one then the outcome IS 50/50 as to whether you pick the right door.   Therefore, the difference between the performance of switching and a random selection of the final 2 doors must be explained.  I noted one important point.  The decision to change door in the original Monty problem was not random.  This was worth exploring.  If you made the decision itself a random choice (I.e. Whether to change or not) the outcome turns out no better than random selection.  I.e. 50/50.  So the performance improvement rests on the decision and therefore must be linked to knowledge.

The key here is that if you KNOW the maths behind this problem then you would switch.  If you did not then your decision would essentially be random.  The performance improvement simply comes from the application of knowledge of Monty Hall problem.  So is the maths itself telling us much? 

To make the problem more obvious lets increase the size of the problem.  Infinite Monty Hall version.  Now there are infinite doors.  Contestant selects one.  The odds of a correct door are therefore 1/infinity.  All we can say about this concept is that the number is very small but not 0.  Monty now opens all the doors apart from one leaving two doors only as per the original problem.  The remaining door now has odds of infinity-1/infinity.  This again is incalculable as infinity is a concept but the number is very large but not 1.

So in summary, the odds of the original door are as close to being 0 as possible without being 0 and the other door as close to 1 (certainty) as possible without being 1.  If you knew  the Monty Hall problem then you would select the highest odds door.  Given this rather extreme view of the problem it is clear that the mathematicians should not be surprised at all that if one is in possession of knowledge (any) about the outcome of an otherwise random event that your success rate will be improved.  That's all Monty Halls shows us in my view.  It is a complex way of stating a rather obvious statement.  You could use dice to show the same thing.  If you know a dice has 2 sixes printed on it then you would do better picking 6 each time.  If you did not know this then you would probably still perform the same as a random selection.

This ramble through the world of numbers raised some other interesting explanatory issues with the maths.  The first is the concept of random selection of a number.  If an event has two outcomes then you have two events going on.  The first is the random selection of two outcomes (in the mind or on computer).  The second is the random outcome itself.   The two random events (not one as is normally discussed) is why the explanation above works.

And that's the itch scratched.  How could the Monty Hall maths change reality, how did the knowledge of the doors alter a physical reality outcome.  It did not and that's because I was missing the fact that there are two events being measured.  The physical reality of the odds of the car behind the door (thanks to the maths guys and Dan for sorting this) and the event of selection.  The knowledge changed the selection from random to informed and that increased the success rate.