This evening, I wrote a letter to my nephew on his 18th birthday. I like to torture him, so I wrote:
Up to this point the world has been nothing but possibilities; you could walk through any door you want to an exciting future. This is true, if by “exciting future” you mean a future in which you struggle your whole life to get by and not despair so much that you finally just give up either via long drop and sudden stop or just by refusing to get out of bed. But the moment you walk through that door, you will hear all those other doors slam shut. And you will be looking at… It’s kind of like Let’s Make a Deal except there are more doors and a llama is behind every one of them.
This brought to mind Monty Hall (who is still alive) and the Monty Hall Problem and, unfortunately, Marilyn vos Savant. I say “unfortunately” because I find her annoying. She is exactly the kind of snarky intellectual that I hate. Here is my biggest complaint: she advertises herself as having the highest IQ in the world, but when anyone questions that claim, she responds that the IQ doesn’t really mean that much. That’s quite true: the IQ is a measure of a certain type of mental functioning; to say it is limited is to be charitable. But vos Savant’s whole career is based upon this claim. If it doesn’t mean much, why is it listed in everything she writes? All of this should not be taken to mean that I think she is stupid—just a PITA self-promoter.
I am grateful to vos Savant for introducing me to the Monty Hall Problem. Here it is: suppose you are on Let’s Make a Deal and there is a new car behind one of the doors and a llama behind each of the other two. You pick door number one. Monty says, “Are you sure you want to pick that door?” And to entice you to change your door, he opens door number three and shows that behind it is a llama. So now you have two doors: one has a car and one has a llama. Should you change to door number two?
This puzzle is counter-intuitive. My first guess (and yours too I bet) is that you shouldn’t change doors, or rather that it doesn’t matter: there is an equal chance of the car being behind each one of the doors. But this is utterly false. Think about it this way: at the beginning, there is 1/3 chance the car is behind door number one; there is a 2/3 chance that it is behind door number two or door number three. So after door number three is taken out of the equation, there is a 2/3 chance that the car will be behind door number two.
You don’t believe me though, do you? That’s okay; I wouldn’t either. But before you embarrass yourself, you should do what slow thinking Frank did: get out a deck of cards. Take three cards and define one of them as the car. Then run through the process. If you are smart (like vos Savant or even me) you will quickly (like after one or two deals) see it (in the religious sense). If you are not so smart, just do it ten or twenty times and you will see that by switching, you will win the car about 70% of the time. Q-E-fucking-D!
You might wonder why switching has this effect. I’m not as smart as Ms. vos Savant (although I’m more fun at parties), but I think I can help. By showing you one of the llama doors, Monty is adding information to the system. When you stick with your original choice, you are not taking advantage of the new information. There is always a 1/3 chance that the car is behind door number one; but there is a 1/3 chance it is behind door number two at the beginning and a 2/3 chance it is behind door number two at the end.
It is not considered ethical to torture people. This is why they invented probability theory. Luckily, I have my nephew to abuse.
Check out the excellent New York Times article/interview with Monty Hall that will explain it all, including some aspects that I have not talked about.