Monty Hall Dilemma: What Is The Probability If You Stay or Switch?

40 years ago, in 1975, the television game show, Let’s Make a Deal, inspired a brain teaser posed in a letter to a magazine called the American Statistician. It became famous when it was again posted in 1990 in another magazine called Parade in a column called Ask Marilyn. 

I posed this question to the  students this week. Students were tasked with explaining it to their parents and testing experiential probability with trials. I hope you have fun with this challenge.

Monty Hall was the game show host on Let’s Make a Deal and would always ask contestants to choose prizes from three doors. INCIDENTALLY, MONTY HALL DIED ON SEPTEMBER 30, 2017 AT THE AGE OF 96. Here is the math variation: suppose Monty says that a car is behind one of three doors and goats are behind two of the other doors. Cars are valuable and goats are not. Monty knows where the car is.

After you choose a door, let’s say, number 1, Monty says that he will open one of the remaining two doors to reveal a goat. Let’s say that he opens door number 2 to reveal a goat. Monty now asks you whether you want to stay with door number 1 or switch to door number 3. 

The question is, “is it advantageous to stay or switch or does it not matter?” What are the mathematical probabilities of your choices?

In 1990, over 10,000 readers of Parade magazine wrote in claiming that it did not matter whether you stay or switch; the probability is 1/2 for each given the two remaining doors. Those who wrote in included 1,000 readers with PhDs. Even the great mathematician, Paul Erdos, did not agree that you should always SWITCH until he was shown a mathematical proof.

We did actual experiments with the students with 3 doors, then with 10 doors and then with 40 doors. It seemed rather odd to them that with 3 doors, switching resulted in winning the car twice as often as staying with your original choice. Before I explain it with more than 3 doors, look at the possibilities when you choose to STAY or SWITCH.

If you choose door number 1 in the three scenarios (the car is behind either door number 1, 2, or 3), you would win the car 1/3 times if you STAY and 2/3 times if you SWITCH. See page 1 of the attached pdf file.

This graphical approach was very convincing to the children but the most persuasive argument is a simulation with 10-100 doors. In an example with 40 doors, I used cups and had them choose the door behind which they believed the car to be. If they chose cup number 1, I would then knock over 38 other cups none of which contained the car. I would only leave their choice (cup number 1) and one other cup (let’s say cup number 27). I would remind them that I know where the car is and ask them if they want to STAY or SWITCH. It became apparently clear that the chances of them being right the first time was, in fact, 1/40 and if they switch to cup number 27 their chances improve to 39/40. In only one experiment did the student pick the car on the first try. When they "correctly" chose to switch in this rare situation, they win a goat but all the children still maintained that switching with the right thing to do. See page 2 of the pdf.

In each class, one student near the end proclaimed the ultimate understanding: “the more doors there are in the Monty Hall problem the probability that you win a car gets higher if you SWITCH and your chances are lower if you STAY.” It is all a matter of statistics. 

During the week, they should try to convince as many people as possible that you should always switch, and even if you are only playing with 3 doors, your chances of winning the car double if you switch. If you are playing with four doors, your chances of winning triple if you switch; 5 doors, quadruple if you switch;10 doors, 9 times if you switch, and so on. They should test theoretical probability against experiential probability by recording several dozen trials for a given number of cups. I hope they will see that the more trials they conduct the closer the experiential trials get to the theoretical probability. For example, if they use 10 cups and conduct 30 trials, if the contestant always switches, the contestant should win the car approximately 27 times out of 30; they should win a goat 3 times out of 30.

There is a script for the students on page 1 of the pdf followed by two pages of graphical explanations with 3 and 40 doors, and then five pages of trials for recording their experiments.

If you still have questions after reading this, contact David Kramer, or better, ask your children.