Vos Savant is wrong...

I've always found the Vos Savant answer to the "Monty Hall" problem troubling and wrong.  Not because I am ignorant of odds and probabilities, but because it does not really match reality and its always posed without a clear set of assumptions.

Here's why...

As far as I can tell this paper (image right) (https://www.jstor.org/stable/2683689) is the first reference to it.  (Though it's apparently a rehash of "The Three Prisoners Problem" appearing in Martin Gardner’s Mathematical Games column in Scientific American in 1959 - no direct link I can find but google "Gardner, M. (1959) Problems involving questions of probability and ambiguity, Scientific American, vol. 201, no. 4, pages 180–182.")

The table in the left column lays things out clearly.

Marylin Vos Savant in Parade magazine's 1990 answer to someone posing this question:

"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?"

Is to switch doors and implicitly you choice of two doors after one is revealed is not 50/50.

Obviously if I stay in the "green room" unable to hear the show and walk out to see only a choice oftwo doors my odds are 50/50.  So the key is what information is conveyed by one door being already open (or at least so it seems).

There's a mostly correct refutation of Vos Vasant's analysis here (Don’t Switch! Why Mathematicians’ Answer to the Monty Hall Problem is Wrong Clive Rix, University of Leicester) write up here: https://cdn.ima.org.uk/wp/wp-content/uploads/2015/08/Dont-Switch-Why-Mathematicians-Answer-to-the-Monty-Hall-Problem-is-Wrong.pdf  By "mostly correct" I mean Rix's math is correct but the assumptions are not quite taken into account correctly.

The key element of this article is in Section 4 where Rix discusses how Vos Savant phrases the problem "the host, who knows what’s behind the doors and will always avoid the one with the prize" (Rix's emphasis). Rix goes on: "Vos Savant has done what all academics do when putting this problem to students (and I do when I am putting it to mine), she has turned a real problem that mathematicians cannot answer because they don’t know what the true probabilities are into one they can answer, by making plausible assumptions. Except that it is not clear in this case how plausible the assumptions are."

I have smelled bullshit for decades.

A recent argument made be work through the issues. Rix addresses some of it in section 4 under Assumptions.  But here I think he makes a mistake.  He writes about assumptions form a mathematicians perspective.

He should be writing from the perspective of "the House" - Monty's employer.

Rix's Assumption 5 is wrong:  Monty works for a show.  The show is part of a network. The show needs to make money for the network.  The money the show makes in advertising more than pays for the prizes (or the prizes are simply provided for free to the show for advertising).  Anyone who owned a business would know this.

Further, the show cannot have all winners (boring) or all losers (also boring).  The show has to be exciting to keep its ratings up.

The show has multiple sponsors competing for product placement on the show.  Can't give away more Ford's than Chevy's, more trips to Hawaii than goats, etc.

So "the House" as three basic options: 

1. Fairly prevent the contestant from winning to the extent possible.
   
   
2. Don't care if the contestant wins or loses (makes the show more exciting).
   
   
3. Allow (or possibly help) the contestant to win.

These options throttle profits and ratings.  The odds always favor the House.

So how Vos Savant phrases this ("the host, who knows what’s behind the doors and will always avoid the one with the prize") hand-waves away reality and replaces it with Selvin's "win/loses" table for only House option #1.

This is a bad business choice for the show.

And, as Rix says, no one can know how the show actually operated or why (in terms of profits, ratings, etc. and there for odds). Hence no one, no matter how smart they are (Vos Savant included) can "know" the true odds.

Having watched the show as a child Monty Hall could easily have offered to simply "buy" the box with the keys in it and not shown what's behind a door.  Or, if the contestant picks a losing box whipped up a huge, artificial frenzy around switching losing boxes.

So my real problem is the fact that Vos Savant creates an environment where you think she is addressing reality but has in fact neutered the problem to the point where its a simple math table tied to only part of reality.

Now take a look at the end of Selvin's "Monty Hall/Contestant" dialog:

Monty, full well knowing he has bosses, ratings and sponsors to manage, presents the odds to the contestant as 1/2.  Why 1/2?  Because only Monty or his bosses know the true configuration of how the contents is set up and why.  My guess is that somewhere there is a professional statistician under NDA that set all this up with the script writers.  Big 1960's TV networks were not manned by fools.

And the odds are 50/50 unless you remember that Vos Savant has quietly clipped out two thirds of the "House's" options.

Now if you look at Wikipedia (https://en.wikipedia.org/wiki/Monty_Hall_problem) you see this:

You'll see that accepting the "standard assumptions" is required to accept Vos Vasant's answer.  Yet there are never clearly provided not vetted.  No one every prefaces the presentation of the problem with "assumptions" because that would make it clear they did not and could not really answer the problem.

Again, anyone who suggests that the you're an idiot if you don't believe in Vos Vasant's answer without providing an agreed upon set of assumptions is the true fool.