The monty hall problem

monty_hall.txt

| Musings on the Monty Hall problem 
|
| Connor Osborn 
| 06/20/15


The problem: 
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?

My intuition: 
Every game boils down to two mystery doors. The door you originally chose and
the door the host didn't reveal. Every game has a goat door that can be
removed. Again you will face two mystery doors, neither of which transparently
reveal a car. It doesn't matter if you switch because your odds are 50/50.

I'm wrong and most people are. You should always switch.

The error in my intuition: 
There are two goats and three doors. Each door is twice as likely to have a
goat than a car, i.e each door has 2/3 probability of hosting a goat. The host
changes this probability in a subtle way. The host will never reveal a goat
behind your door. The probability of your door having a goat doesn't change.
However, the host always reveals a goat from the other two. The probability of
those doors does change!

Initial odds(goat):
*------------------------------------*
|             6/3, 2 goats           |
 *---------*  *---------------------* 
 |   2/3   |  |         4/3         | 
  *-------*    *-------*   *-------*  
  |       |    |       |   |       |  
  |  2/3  |    |  2/3  |   |  2/3  |  
  |o      |    |o      |   |o      |  
  |       |    |       |   |       |  
  |       |    |       |   |       |  
  *-------*    *-------*   *-------*  
   Door 1       Door 2      Door 3

Host reveals goat!

Final odds:
*------------------------------------*
|             6/3, 2 goats           |
 *---------*  *---------------------* 
 |   2/3   |  |         4/3         | 
  *-------*    *-------*   *-------*   
  |       |    |       |   |       |   
  |  2/3  |    |  1/3  |   |  3/3  |   
  |o      |    |o      |   |  (__) |   
  |       |    |       |   |  o o\ |   
  |       |    |       |   | ('')  |   
  *-------*    *-------*   *-------*   
   Door 1       Door 2      Door 3

Door 1 is twice as likely to have a goat!