Simulation of a problem inspired by the classic game show Let's Make a Deal
Named after the charismatic Canadian host, producer and co-creator of the long-running American game show.
Each day, the show would culminate with the "Big Deal" where a significant prize, such as a new car, would
be concealed behind one of three doors. Up to three contestants were given the opportunity to choose a door
and keep whatever was behind it (trading in the prizes they had won earlier in the show).
The Monty Hall problem is
inspired by this game, but with several twists. First, there is only one player. Second, there is only
one prize (nothing behind the other two doors). The third, and most significant, feature is that after the
contestant picks his door, Monty (who knows where the prize is) opens one of the other doors to show that
there is no prize there, then he offers to let the contestant change his mind and choose the other remaining
door. The nature of the problem is this: are the odds of winning better with the other door? Should the
contestant switch?
Here, you get to be the contestant. You pick a door. Monty will show you an empty door. Then you decide
whether to stick with the door you originally chose, or switch. Do it a bunch of times to get a feel for which
strategy is better...
Click on the door you want to choose.
This should be invisible
Monty now opens one of his two doors, being careful not
to reveal the prize...
The goat represents an empty door—Monty opened this door
knowing that the prize was not there.
Clearly, the prize has to be behind one of the two remaining doors.
You started with a 1 in 3 chance. What are the odds now? You can choose to stick with
your original guess, or trade for the other door...
Interpreting the results
Almost everyone who hears the Monty Hall problem recognizes that, when they first choose a door, there is a 1 in 3
(i.e., 0.33, 0r 33%) probability that they picked the prize door. But when Monty opens one of his empty doors, reducing
the possible locations of the prize to two doors, there is a strong tendency to believe that the odds have somehow
changed to 50:50. In fact they have not changed: when you first choose a door, there is a 2/3 probability that you
are wrong—that the prize is in one of the other two doors. And no matter where the prize is, at least one of
those other two doors must be empty. So Monty showing you a door that he knows to be empty does not change the odds.
There is still a 1/3 chance that you chose the right door and a 2/3 chance that you chose the wrong door.
So if you can change your guess, you will improve the probabilty of winning, from 1/3 to 2/3.*
If you reload this page and play the game again a number of times, you should be able to prove to
yourself that trading wins twice as often as sticking.
*Naturally, this would not be true if Monty was trying to trick you, and only offered to trade if
you had picked the winning door. There is no such manipulation here—we told you from the start that this offer
would be made, regardless of your choice of door.
