The Monty Hall problem is one of my favorite math problems. It has that perfect mix of math and philosophy that is right up my alley. The problem is relatively simple, but it says so much about how we deal with seemingly obvious outcomes. It challenges our basic assumptions and makes us question our beliefs. Also, you know, it’s math-y.
The Monty Hall Problem
The Monty Hall problem rose to prominence in an entertaining 1990 Parade column. Wikipedia also has a really good breakdown of the problem. I recommend you read both, but I am going to briefly (as much as possible) go over the basics:
You are on a game show. Three doors stand before you. Behind one is the super awesome way cool grand prize. Behind the other two is nothing. The host asks you to pick a door. You do so, because it’s a game show and all. The host then opens a door that is not the grand prize. There are now two closed doors, one of which was your initial selection. Before the final reveal, the host offers you the option of switching doors. What do you do?
Most people with a basic understanding of math would say that there is no reason to switch doors. Every door has a 1/3 chance of being the grand prize, so switching doesn’t matter. But given the context of this problem, you probably realize it isn’t as straight forward as that.
The answer is that you should switch doors. By switching doors, you have a 2/3 chance of winning the grand prize. If you stay with your original door, you only have a 1/3 chance of winning. But how is that possible?
The key point in the Monty Hall problem is when the host opens a door. It is important to understand that the host is not opening a random door. The host is opening a door with a condition, specifically a door that is not the grand prize. This is probably the “trickiest” part of the Monty Hall problem, because we don’t naturally make this distinction as being note-worthy. Who cares if it isn’t the grand prize? One out of three is one out of three, right?
In this case no. Let’s walk through it step by step. You pick a door, we’ll say door number 1. At this point in time, it is true that you have a 1/3 chance of selecting the grand prize door. Now, the host reveals a door that is not the grand prize, we’ll say door number 3. Well now things have certainly changed. Now you know that door number 3 isn’t the grand prize. Here are the possible outcomes now:
Door 1 is the grand prize. You win!
Door 2 is the grand prize. You lose. 🙁
Still looks like a 50/50 shot, right? So switching is still pointless. But again, this is deceiving. You still have to factor in Door 3. Think of it this way. If Door 3 had the grand prize, Door 2 would have been revealed to be empty. Then if you switch from Door 1 to Door 3, you win. And in our scenario above, if Door 2 is the grand prize and you switch from Door 1 to Door 2, you win. In fact, the only way you can win with Door 1 is if you just happened to pick the winning door at the beginning.
To state it more clearly: You only win 1/3 of the time if you do not switch doors. That is when you initially pick the grand prize door. But if you switch, you win 2/3 of the time. That is because you would win if either of the other two doors are the grand prize, since one is eliminated from consideration.
At this point, you may or may not “get it”. That is just as much on me as it is you. It can be difficult to explain and if you are anything like me, you need it explained in just the right way to understand. But some people, it seems, simply reject the answer despite given evidence.
But but but! (Clinging onto basic assumptions)
For myself, the most interesting aspect of the Monty Hall problem comes from people refusing to accept the answer. Humans are hardwired to assume. We generalize. And this is a good thing. Our ability to generalize most likely played a key role in our evolutionary survival. Just imagine if we didn’t make assumptions. “Oh hey there Mr. Lion. I know that all the lions I have seen in the past have devoured my friends, but I’m not going to assume anything based on the actions of those lions. Let’s see if we can be friends.”
Generalization in modern society gets us into a whole world of problems (racism anyone?), but it’s good to recognize we generalize for a reason. We simply don’t have the brain power to approach every new situation from scratch. Nor should we have to. But we do need to recognize when our assumptions are being challenged and be able to change them.
The Monty Hall problem is such a perfect example of this phenomenon. Even I still questioned it after first hearing the explanation. In some ways, that the problem is so simple makes it even harder to accept. “What do you mean it isn’t 1/3!? Of course it is. It has to be. If it isn’t 1/3, then MY LIFE IS A LIE.”
When we dig down to our fundamental beliefs and challenge them, there is likely to be resistance. That is true in all aspects of life. The Monty Hall problem illustrates this. But I argue that challenging our own believes is itself fundamental to life. Progress is not made through acceptance of the norm, but by questioning it. We must be aware of our assumptions and our generalizations. We must know when they are overriding our logic.
And most of all, we must know when to switch doors.