What Is Behind a Closed Door?

Years ago, Monty Hall was the host of the television game show ‘Let’s Make a Deal,’ One of the contests on the show started with three closed doors. Behind one of the doors was a car.  The contestants would pick one of the doors. Monty would then open one of the other two doors, showing the contestant that the car wasn’t there.  He’d ask the contestant if they wanted to switch doors.

Intuitively you would think it doesn’t matter. It’s a fifty-fifty chance, right? Heads or tails, black or red, right or wrong, right? It’s obvious! The thing is, it’s not that simple.

In mid-1991 someone sent the problem to Parade Magazine’s Ask Marilyn column, since called the Monty Hall Problem. Marilyn Von Savant, the smartest person in the world at that time, according to The Guinness Book of World Records, answered it, saying, “Yes, you should switch. Youd have a 2/3 better chance of winning if you switched.”

She received over 10,000 letters telling her that she was wrong — the odds were 50-50. She even received hundreds of letters from math teachers, physics teachers, and even university math professors.  People really enjoyed correcting the smartest person in the world.

The thing is she was right, and a simple computer program will verify it. How can this be? It’s so intuitive.

It’s easier to see with a larger set. Let’s say there’s a 10’ by 10’ table with a hundred white envelopes arranged in 10 rows by 10 columns. In one of the envelopes is a check for a million dollars. I let you pick one of the envelopes and hold it.  If I offered you the chance to trade your one envelope for the other 99 you’d be a fool to not take it. Right?

Now I open 98 envelopes leaving one. I ask you again if you would like to trade. You might think that it’s now a 50-50 chance. But what has really changed? What have I really added to the problem? You knew that there were 99 empty envelopes. I knew there were 99 empty envelopes. I haven’t showed you anything you didn’t already know. You see, I know where the check is. All I did was clear away 98 envelopes I knew were empty. The last envelope on the table is either empty or the check is in it. I know which. There is still a 99% chance that the check is on the table.

Now consider the same example. This time instead of knowing where the check is, I’m going to make random guesses. You’ve made your choice and I have 99 envelopes to choose from. Assuming the check is in one of those 99 envelopes, my odds are 1 in 99. I pick and it’s empty. Now my odds are 1 in 98 Another empty envelope makes it 1 in 97. I keep going and soon there are only two envelopes left. Now my odds are down to 50-50. Remember you still have your envelope.  I pick one of the two and it’s empty.

Now if I were certain that the check was in one of the 99 envelopes left after you made your pick, the chance that the check is in the last envelope is 100%. In fact, there is no chance at all that it wouldn’t be there.

Now, we know that to start with, the check was either in your envelope or on the table, in one of 99 envelopes. It’s still either in your envelope or on the table and since it’s clearly not on the surface of the table if it’s still on the table it must be in the envelope.  I made 98 random guesses and that changed the odds with each guess. I added new information to problem. Every time I picked an empty envelope it told us something we didn’t know. By taking 98 chances the odds improved from 99 to 1 to 50-50.  

This is just as true for any number of choices. There were three doors. Monty knew which door had a car hiding behind it. He never told the contestant that. It seemed Monty was making a random choice. So, when Monty opened door without the car, the contestant intuitively felt the odds had been changed to 50-50. And one door was as good as the other.

By Tim Shively, May 22, 2019



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