Goats and Cars: Try It!

Not convinced by theoretical arguments? There's nothing like trying it experimentally. Direct experience is most convincing.

Equipment

  1. Grab a whole bunch of quarters and pennies. These represent the two goats and the car, but in this experiment we're going to play the game many times, and keep our prizes from each, so we'll need a good supply of tokens to represent cars and goats. Maybe you have a bunch of toy cars? Go ahead and use them. If you are farmer or otherwise rurally inclined, you may use real goats, however we have found this to be cumbersome.
  2. Find three small boxes, or envelopes or books or any other object convenient for hiding the coins in or under. (We shall refer to these objects henceforth as "boxes", no matter whatever you're really using.) Three of them, to represent the three doors.
  3. Also get your hands on two bowls or cups, or if you're so poverty-stricken that this is not possible, just draw a couple of circles on a sheet of paper. Label these containers "SWITCH" and "STAY." These will hold the prizes you win, goats and cars.
  4. If you've been reading this web site a while and still don't believe the answer to the puzzle, then you're probably too confused to trust yourself at hiding coins. Find someone to hide the coins for you. That person will be the "host."

The Procedure

Follow these steps to simulate the game.
  1. The Host takes two pennies and one quarter from the supply pile and hides each coin in a box. Each box has one coin, and each coin has a box. (No peeking!)
  2. As the contestant, pick one of the boxes. This is your first choice. But do not reveal its contents yet! (If you're host is snickering at you for making the silly choice you did, tell him/her to stop it!)
  3. Your Host, knowing which coins are hiding where, now reveals a penny from one of the two boxes that you didn't pick.
  4. Now pick a door again - decide if you want to stay with your first choice or switch to the other box. Follow your whim. For a more scientific approach, flip a coin (if you have plenty not yet in use) to decide.
  5. Open your chosen box. If you had stayed with your first choice, put the coin from that box into the STAY bowl, whether it's a quarter (pretending to be a car) or a penny (pretending to be a goat). If you had switched doors, put the prize coin in the SWITCH bowl. (The other coins can be returned to the original supply pile.)
  6. Repeat the whole thing from Step One, over and over, a whole bunch of times. Repeat until you use up all your money. Repeat until you're too tired to stand up. (It's just like Las Vegas!)

It is important to repeat the game many times, with the Host hiding three coins randomly each time. Repeat at least twenty times, but the more the better.

The Results

Look at the contents of both prize containers - the STAY and the SWITCH prizes you won. Which stash do you like better?

Conclusion

Just because there are two options available, it does not follow that the probabilities of some outcome associated with them are 50% and 50%.

Legend has it that Monty Hall himself had heard of this problem, and in his home did an experiment not unlike that described above. This proved wrong those who insisted that switching doors was a 50-50 situation - and that Marilyn Vos Savant's math is correct - but Mr. Hall, being an old pro in the business, pointed out a subtle feature in the wording of the original problem. Read all about it at John Tierney's well-researched article for the New York Times. about the "Henry James Treatment".
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© Daren Scot Wilson, website
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Contact the author at darenw@comcast.net