## Mathematicians miss the moral of the Monty Hall problem

Let me take for granted
1. The usual analysis of the Monty Hall problem is correct.
2. Most people, and many mathematicians, find the answer unintuitive at first sight; many people, and some mathematicians, still find it unintuitive after seeing the explanation.
But what is the moral of the story? I think that mathematicians view the moral as follows.
• (*) This is a typical instance of how untrained intuition about probability gets it wrong and how using the formal setup of mathematical probability gets it right.
If you don't believe (*) then stop reading -- sorry, I've wasted your time. For those still reading, I will argue
• Assertion (*) is wrong.
To stop you getting annoyed later I should confess up front that I'm playing a bit of a trick. Remove the word "typical" from (*) and it becomes a rephrasing of (1,2) which is perfectly correct: it is an instance of ..... . So what I'm arguing about is the word typical.

Let me make an analogy with Rubik's_Cube. What makes that fascinating, and hard to solve as a puzzle, it that it's different from familiar physical objects. It's an artifact, designed to be different and challenging. Analogously, a key feature of the Monty Hall problem is that it's quite different from other "decision under uncertainty" problems; one cannot imagine any closely similar problem arising, outside the context of a game with rules invented to make it challenging.

It is true that the Monte Hall problem is one of a standard list of mathematically elementary toy problems that one easily gets right by using the formal setup; but one can create a less standard list of slightly more real-world problems for which the formal setup leads you astray: here is one example. The basic point (discussed at greater length here: xxx to be written) is that in the real world we get information that has been selected in ways we can't analyze. In Monty Hall there are only three equally likely initial possibilities and an agent acting according to a known deterministic rule, so we can do the analysis. Can you think of any real-world story, outside of an invented game, with this simple a structure?