Creation: July 05 2017

Modified: February 05 2022# The Monty Hall Problem

## Introduction

The Monty Hall problem is a counter-intuitive statistics puzzle loosely
based on the American television game show *Let's Make a Deal* and named
after its original host, Manty Hall.

The Game:

- There are three doors, behind which are two goats and a car;
- You randomly pick a door;
- Monty, examines the other two doors and always opens one of them with
a goat.

Do you stick with the original guess or switch to the other unopened door?

Surprisingly, the odds to win the car ar 1/3 if you stick with the original
one but goes to 2/3 if you switch the door.

The game is about re-evaluating your decisions as new information emerges.

## Evident Proof

Initially you can pick up one of three doors. Only one with the car.
Assume that the door with the car is W and the other two are B and C.
Let's enumerate the possible cases:

- You choose W. Monty removes B or C.
If you stick with the original choice you win, else you loose.
- You choose B. Monty removes C.
If you stick with the original choice you loose, else you win.
- You choose C. Monty removes B.
If you stick with the original choice you loose, else you win.

As you can see from the evidence:
If you don't change your choice you loose 2/3 of the times.
If you change your choice you win 2/3 of the times.

## Explanation

The problem can be generalized to N doors. Where only one has a car behind.

If you stick with the original choice the odds to win are just 1/N, but if
you change your mind, after that Monty has removed N-2 goats, then the odds
to win magically becomes (N-1)/N. Pretty impressive.

To better understand why this works, another, alternative, approach to the
problem is needed:
If you change your initial choice is like if, initially, you are not trying
to pick the door with the car but a door with a goat, which has probability
(N-1)/N). Then Monty will always make you the favour of removing the other
N-2 bad cases and what is left is, eventually, the fortunate case.

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