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.

In short, although it may contradict "common sense", The Contestant will double their chance of winning by switching doors.


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