Monty Hall Dilemma

The Monty Hall Dilemma was discussed in the popular "Ask Marylin" question-and-answer column of the Parade magazine. Details can also be found in the "Power of Logical Thinking" by Marylin vos Savant, St. Martin's Press, 1996.

Marylin received the following question:

Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors?
Craig. F. Whitaker
Columbia, MD

Marylin's response caused an avalanche of correspondence, mostly from people who would not accept her solution. Several iterations of correspondence ensued. Eventually, she issued a call to Math teachers among her readers to organize experiments and send her the charts. Some readers with access to computers ran computer simulations. At long last the truth was established and accepted.

Below is one simulation you may try on your computer. For simplicity, I do not hide goats behind the doors. There is only one 'abstract' prize. You may either hit on the right door or miss it. You make your selection by pressing small round buttons below input controls that substitute for the doors. Down below other controls update experiment statistics even as you progress.

Approximately every second the program clears 'door' controls and is waiting for your selection. Before you start, set up a desirable total number of experiments. With every selection it will decrease by 1.

Two controversial solutions are given after the puzzle. Which is the right one?

Doors When you select a door and miss the prize the word 'selected' appears in the control. The one open by the host will display the word 'open'. If you guess correctly the word 'prize' will be displayed in the corresponding control.
Use these radio buttons to make your selection.
#Wins#Losses
If you switch
If you do not switch

Number of hits
Trials to go

Solution #1

There is a 1/3 chance that you'll hit the prize door, and a 2/3 chance that you'll miss the prize. If you do not switch, 1/3 is your probability to get the prize. However, if you missed (and this with the probability of 2/3) then the prize is behind one of the remaining two doors. Furthermore, of these two, the host will open the empty one, leaving the prize door closed. Therefore, if you miss and then switch, you are certain to get the prize. Summing up, if you do not switch your chance of winning is 1/3 whereas if you do switch your chance of winning is 2/3.

Solution #2

After the host opened one door, two remained closed with equal probabilities of having the prize behind them. Therefore, regardless of whether you switch or not you have a 50-50 chance(i.e, with probabilities 1/2) to hit or miss the prize door.

Remark

The above simulation tool have the virtue of being quite suggestive - three quantities, viz.,

  • the number of hits
  • the number of wins with no switching
  • the number of losses with switching

are all equal. It's really better to see once...

Multi-Stage Monty Hall Dilemma

"In the three-door Monty Hall Dilemma, there are two stages to the decision, the initial pick followed by the decision to stick with it or switch to the only other remaining alternative after the host has shown an incorrect door. An intriguing extension of the basic Monty Hall Dilemma has been provided by M. Bhaskara Rao of the Department of Statistics at the North Dakota University. He analyzed what happens when the dilemma is expanded beyond the two stages. The number of stages can be as many as the number of doors minus one.

"Suppose there are four doors, one of which is a winner. The host says:

"You point to one of the doors, and then I will open one of the other non-winners. Then you decide whether to stick with your original pick or switch to one of the remaining doors. Then I will open another (other than the current pick) non-winner. You will then make your final decision by sticking with the door picked on the previous decision or by switching to the only other remaining door.

"Now there are three stages, and the four different strategies can be summarized as follows:

Stage123Probability of winning
PickStickStick.250
PickSwitchStick.375
PickStickSwitch.750
PickSwitchSwitch.625

"People who accept the correctness of the 2/3 solution in the basic Monty Hall Dilemma might assume that one does best by switching in both Stage 2 and Stage 3. However, as shown here, the counter-intuitive solution to the three-stage Monty Hall Dilemma is to stick in Stage 2 and to switch in Stage 3. These remarkable probabilities were published by Rao in the American Statistician. The underlying principle is that in a multi-stage Monty Hall Dilemma, one should stick with one's initial hunch until the very last chance and then switch."

Three Shell Game

Martin Gardner in his Aha! Gotcha describes the following variant:

Operator: Step right up, folks. See if you can guess which shell the pea is under. Double your money if you win.

After playing the game a while, Mr. Mark decided he couldn't win more than once out of three.

Operator: Don't leave, Mac. I'll give you a break. Pick any shell. I'll turn over an empty one. Then the pea has to be under one of the other two, so your chances of winning go way up.

Poor Mr. Mark went broke fast. He did not realize that turning an empty shell had no effect on his chances. Do you see why?

Comment

The problem is actually the same but looked at from a different perspective. Since Mr. Mark has made his choice no Operator's action can change his chances. So, to me at least, the Shell Game makes it pretty obvious that unless you switch in the Monty Hall Dilemma (i.e. if you play the Shell Game), you chances remain 1 to 3. However, if you switch, you select one door out of two.

References

  1. A.K.Dewdney, 200% of Nothing, John Wiley & Sons, Inc., 1993
  2. Martin Gardner, aha! Gotcha. Paradoxes to puzzle and delight, Freeman & Co, NY, 1982
  3. Marylin vos Savant, The Power of Logical Thinking, St. Martin's Press, NY 1996

On Internet

  1. http://math.rice.edu/~ddonovan/montyurl.html">The WWW Tackles The Monty Hall Problem

Copyright © 1996-2002 Alexander Bogomolny