## Mathematicians miss the moral of the Monty Hall problem

Let me take for granted
- The usual analysis of the
Monty Hall problem is
correct.
- 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
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?