Let's Make a Deal: Using Monty Hall to Address Educational Issues

One of my favorite CPs here at Associated Content is Brian McCormick. He doesn't write as much for the site as he used to, but he recently composed an article about how flawed our educational system is because of its emphasis on memorization. He wrote:

"Memorization is no longer the most important skill. Possessing information offers no advantage.

Instead, it is the ability to use information which differentiates successful people. This requires creativity, problem-solving, imagination, analytical skills, abstract thinking and more. These are the skills we need to teach in today's world."

I was thinking about his article when I came across a discussion about probability and the old "Let's Make a Deal" game show which I think illustrates his point pretty good. In the game show, there was a segment in which the contestant had his choice of three doors. Behind one door was a car and behind the other two doors was something useless, like a cow.

The contestant would pick a door and then the host, Monty Hall, would display what was behind one of the two doors that the contestant did not pick. Obviously, Monty would show a cow and then he would ask the contestant if he wanted to switch his answer from his original choice to the other remaining door.

Now, when I watched the show as a kid I never understood why they asked this question and why anyone would change their answer. Later on, as I became more interested in math, I understood that the reason they gave you a choice was to improve your odds at winning the car. If you think the problem through logically, it becomes obvious.

Yet there are people, smart people, who to this day still don't understand why it is in your best interest to switch doors in this example. To me, this is an indication of our flawed educational system.

We started with three doors, so the contestant has a 1-in-3 chance of winning the car. After Monty reveals a door that has a cow, we are left with two doors - the one originally picked and one other. At this point, Monty would offer the contestant the chance to switch.

On the surface, it seems that there are two doors left and each one has a 50-50 chance of having the car. And if someone entered the show at this point, perhaps that would be the case. But because our contestant has been there from the start, he has additional knowledge.

At the beginning of the game, our contestant realizes that he has a 33 percent chance of having the door with the car and there was a 67 percent chance that it was behind one of the other two doors. Just because Monty revealed a door that did not have the car, does not change the fact that the contestant's door gave him a 33 percent chance at winning.

Monty revealing a door that did not have the car simply switches the 67 percent probability from the two doors into one single door - the door that the contestant did not pick. By switching, the contestant is doubling his odds at winning the car.

I can sense through cyberspace that some of you are still not convinced.

Let's say that we play the game three times. Each time the contestant picks door number one. And each time the car is behind a different door. One time it is door number one, one time it is door number two and one time it is door number three.

In the first game, the car is behind door number one. The contestant picked door number one and Monty revealed that there was a cow behind door number two. He offers the contestant a chance to switch his pick. The contestant does switch and the reveal shows the car behind door number one. The end result is no car for our contestant.

In the second game, the car is behind door number two. The contestant picked door number one and Monty revealed the cow behind door number three. The contestant makes the switch when offered and the end result is a car for our contestant.

In the third game, the car is behind door number three. The contestant picked door number one and Monty revealed the cow behind door number two. The contestant makes the switch when offered and the end result is a car for our contestant.

By switching, he ends up with the car twice. If he kept his original pick, he would have wound up with the car only one time.

As a seven-year old, I understood that our contestant had a one-in-three chance of winning the car at the beginning of the game and I assumed that after the reveal he had a 50-50 shot. I may not have been able to express that in words but that was the thought process in my head.

And as McCormick pointed out in his article, we are teaching kids to memorize the probabilities of the various states in the game but not giving them the skills to solve the problem, or in the "Let's Make a Deal" case, at least to stack the odds in their favor.

So, what's the solution? No multiple choice or fill in the blank or matching questions on exams. Make them all word problems. We would be better off having kids take exams with five word problems than with 50 multiple choice questions. There would be a lot more thinking involved that way. And a lot more people winning cars on "Let's Make a Deal".

Meanwhile, I am trying to work on another side of the issue with my son. We are big into the game "Rock, Paper, Scissors" right now. While there is not a whole lot of thinking involved in that game when we play, there are still outcomes and probabilities at work. And unlike me in the early 70s, my son can express that he has a 1-in-3 shot of winning.

Because articulating ideas and concepts is a strong building block in problem solving.