Figure 1

Here's a problem that makes the round every few years,
and each time, it is hotly debated.
You are on a game show. You are presented with a
choice of 3 doors: behind one is a luxury car, and
behind the other two are nothing.
The host asks you pick one of the doors.
After you do this, as part of the game he
opens one unpicked doors which he knows is empty.
There are now only the door you picked and one remaining
door which are unopened. You are asked if you would
like to switch your choice. Should you switch?
Presentation Suggestions:
Another version of this problem uses cards
in a game called "threecard monte",
often played by scam artists on the streets of New York
who prey on easilyduped tourists.
The Math Behind the Fact:
The Monty Hall Problem, or Monty Hall Paradox, as it is known, is named after the host of the popular game show "Let's Make a Deal" in the 1960's and 70's, who presented contestants with exactly this scenario.
The answer is YES, you should switch, because the
probability that you will find the car by doing so is
2/3. This is because the probability that you picked
the correct door in the first place does not change;
it is still 1/3, regardless of the game show host's actions.
Many people are fooled into thinking that once
one of the doors is eliminated that the
probability between the remaining two doors is now
5050, but this is incorrect.
There are many ways to expose the fallacy; here's one
heuristic argument. Suppose you play this game 600 times.
About 200 times you will
pick the right door at the start. Yay!
About 400 times you will not.
But the game show host will never open the
door with the car behind it, so each of those 400 times
the car is behind the unopened door that you did not
originally pick. So 400 out of 600 times
you should switch, i.e., 2/3 of the time.
Another way to see this is to examine an extreme case.
Suppose you play this game with a deck of 52 cards,
trying to pick the ace of spades, and you pick one card.
Now suppose that of the remaining 51 cards, the dealer
turns over 50 cards which he knows are not the
ace of spades. This leaves one unturned card, aside
from the one you picked.
Should you switch? Of course!
The probability that you picked the correct card to
begin with is 1/52, and the probability that it is in
the other 51 cards is 51/52. Neither of these
probabilities are changed by the dealer's actions, since
he knows the cards and will never turn over the
ace of spades. You should therefore switch your choice.
How to Cite this Page:
Su, Francis E., et al. "Monty Hall Problem."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
