Monty Hall: The Solution

1. For the moment, we will make the simplifying assumption that The Contestant always begins by choosing Door #1. (As we will see, this assumption does not affect the answer.)

2. First we will examine all the possible games in which The Contestant does not trade doors when given a choice.

   As we will see this will ensure that the contestant will win 1 out of 3 times.

   Now, there are three possible games:

   • Possible Game 1:
     The prize was originally behind Door #1:
     

     The Host then opens one of the other doors (which doesn't matter):
     

     Since The Contestant does not switch doors when given the choice:
     

     ==>YOU WIN!!<==

   • Possible Game 2:
     The prize was originally behind Door #2:
     

     The Host then opens Door #3.
     

     Since The Contestant does not switch doors when given the choice:
     

     ==>YOU LOSE!!<==

   • Possible Game 3:
     The prize was originally behind Door #3:
     

     The Host then opens Door #2.
     

     Since The Contestant does not switch doors when given the choice:
     

     ==>YOU LOSE!!<==

3. Summarizing we have the three cases: Game 1: ==>YOU WIN!!
   Game 2: ==>YOU LOSE!!
   Game 3: ==>YOU LOSE!! As we can see: If The Contestant does not change doors he will win one of the three possible games.

   That is, in this case --if The Contestant does not change doors-- the chance of winning is 1/3.

4. Now, we will examine all the possible games in which The Contestant does trade doors when given a choice.

   As we will see this time, the contestant will win 2 out of 3 times.

   As before, in this case there are three possible games:

   • Possible Game 1:
     The prize was originally behind Door #1:
     

     The Host then opens one of the other doors (which doesn't matter):
     

     The Contestant now switches to the remaining unopened door:
     

     The resulting situation is:
     

     ==>YOU LOSE!!<==

   • Possible Game 2:
     The prize was originally behind Door #2:
     

     The Host then opens Door #3.
     

     The Contestant now switches to the remaining unopened door:
     

     The resulting situation is:
     

     ==>YOU WIN!!<==

   • Possible Game 3:
     The prize was originally behind Door #3:
     

     The Host then opens Door #2.
     

     The Contestant now switches to the remaining unopened door:
     

     The resulting situation is:
     

     ==>YOU WIN!!<==

5. Again, summarizing we have three cases: Game 1: ==>YOU LOSE!!
   Game 2: ==>YOU WIN!!
   Game 3: ==>YOU WIN!! As we can see: If The Contestant does change doors he will win two of the three possible games.

   That is, in this case --if the Contestant does change doors-- the chance of winning is 2/3.

This, then, is the final answer.

A Contestant who does not switch doors (i.e. who keeps the originally chosen door) has only a 1/3 chance of winning.

A Contestant who switches doors, however, has a 2/3 chance of winning.