# The Monty Hall Problem - a state-space explanation

 Here is yet another explanation of the answer. It requires tables and color. Some other pages The WWW Tackles The Monty Hall Problem Monty Hall Problem Three Door Puzzle Monty Hall Problem

There are 3 doors
 ? ? ?
one of which has a Prize.
So the universe is in one of 3 possible states
state1
 P - -
state2
 - P -
state3
 - - P
Now, you choose a door.
 ? ? ?
Looking inside, we see P in just one of the 3 possible states.
state1
 P - -
state2
 - P -
state3
 - - P
If the universe is in state1, we get the P.
But if it is in state2 or state3, we don't.
Our chances are 1/3 for "P", and 2/3 for "-".
Now "Monty" takes the other two doors
state1
 P - -
state2
 - P -
state3
 - - P
and opens one without the prize
state1
 P - -
state2
 - P -
state3
 - - P

Now which door do you want, red or green?
state1
 P -
state2
 - P
state3
 - P
In green, we get two P's!
If the universe is in state2 or state3, we get the P.
Only if it is in state1, do we lose.
So our chances are 2/3 for "P", and 1/3 for "-".

 As you know, a 2/3 chance of getting the Prize is better than a 1/3 chance, so we tell "Monty" we've switched our choice to green.

 Comments encouraged. - Mitchell N Charity

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