For those like myself who were unaware, Let’s Make A Deal is a game show that first aired in 1963, hosted by Monty Hall. It actually is still on air today with Wayne Brady as a host. The show consists of audience members being called on stage to make deals and trades with the host and their assistants. In any case, Let’s Make A Deal is also the source of a fascinating mathematical problem that, despite its simplicity, most people including myself get wrong.
I hope you agree that it’s a fun thought experiment!
The Let’s Make A Deal dilemma
This dilemma, also known as the Monty Hall problem, first entered the public zeitgeist in 1990 in the form of a question asked in the Ask Marilyn column of Parade magazine. The titular Marilyn was Marilyn von Savant, a woman with a reported IQ of 228.
The question was posed as such:
Suppose the contestants on a game show are given a choice of three doors: Behind one door is a car and behind the others is nothing. After the contestant picks a door, the host, who knows what’s behind all doors, opens one of the unchosen doors, revealing nothing. He then says to the contestant, “Do you want to switch to the other unopened door?” Is it in the contestant’s advantage to make the switch?
I encourage you to think about this for a second. What’s the answer? Yes? No? And more importantly, why?
I first found this dilemma when reading The Drunkard’s Walk: How Randomness Rules Our Lives by Leonard Mlodinow. And it drove me nuts. It’s a simple question but I couldn’t quite think through an answer that made sense in the 5 plus minutes that I gave myself before proceeding in the book.
The traditional thinking
It’s simple right?
Two doors remain available after the host opens one to reveal nothing behind it. Behind one of the remaining doors is a car. Behind the other is nothing.
Your chances of ending up with the car, whether you switch or not, are 50/50. Easy.
The problem, however, was that Marilyn answered in her column that the contestant would be better off the switch.
And people freaked out!
Phd’s wrote in literally telling Marilyn that she blew it. Everyone was mad. How could such a “smart” person not realize something so simple.
However, it turns out that everyone but Marilyn was actually making the same very basic mistake. Because our brains are just not that well wired to think about probability and chance.
The right way of thinking
The simple mistake that most of us make is that we assume this “Let’s Make A Deal dilemma” deals with three equally probable outcomes. If that were the case, then the answer would be 50/50 and the contestant would be no better switching doors than sticking with their current selection.
But that is not actually the situation we are in. Because of one oft-ignored factor: the host.
The host is not selecting one door to open at random. In fact, the host already knows what is behind all of the doors. So, he or she will open a door that they already know does not contain the car.
As it turns out, awareness of this fact will lead us to the correct, and shockingly simple answer
Let’s examine this from the perspective of the contestant.
I select one door hoping that a brand new “doctor car” will be behind it. At this point, I have a 1/3 chance of selecting the correct door. As Mlodinow puts it in his book, we will label this the “Lucky Guess” scenario.
If we are in this Lucky Guess scenario, the host will then open one of the two remaining doors which will have nothing behind it. We then face a choice, stick with our original door and win a car. Or switch doors and lose. In this scenario, which will occur with a 1:3 probability, we are better off staying with our original pick.
However, there is a 2/3 chance that our original pick was a door with nothing behind it. In this “Wrong Guess” scenario, the host will now open one of the two remaining doors. One of the remaining doors has nothing behind it, the other has the winning car. Of course, the host knows which is which so he or she opens the door with nothing behind it. As things now stand, we are better served to switch our choice which will result in us winning a brand new car!
The key now is to realize that we have a twofold greater chance of finding ourselves in the “Wrong Guess” scenario than the “Lucky Guess” scenario. Therefore, as the contestant, I give myself 2x greater odds of winning by actually switching doors in this Let’s Make A Deal dilemma.
In fact, when analyzing outcomes of the show in this exact scenario, contestants who switched doors won about two times more than those who stuck with their original pick!
So, where doe this leave us?
I think there are 3 main takeaways from this post.
- If you are ever on a game show like this, switch the damn doors,
- It’s important to recognize that we really do not do a good job thinking about probabilities and chance in our lives. Humans are way to quick to assign cause and correlation where none actually exists. So, keep an open mind about things, and
- Yes, this applies to the stock market and investing. It’s why active stock picking doesn’t work even though sometimes it feels like it does…
What do you think? Have you heard of this problem before? Did you figure it out? How does randomness really rule our lives? Let me know in the comments below!
