Here is one of my favorite math puzzles – commonly known as the “Monty Hall Goat Trick”. It’s a wonderfully delicious problem that plays with your head. It has a psychological component that makes even the brightest of people swear that the obvious answer is the correct one, but it’s not!
The game was a regular feature on “Let’s Make A Deal” – a strangely popular show back in the 70’s with legendary game show host Monty Hall. Pop culture at its finest (well, along with “The Gong Show”, of course). During the show, some very lucky (and usually crazily dressed) person was selected from the studio audience to play the game, and had the chance to win A BRAND NEW CAR! (this became Monty Hall’s catchphrase).
The puzzle has been discussed on hundreds of website, so it’s nothing new, but it's still one of my favorites. Stay with me if you haven’t run across this problem before …
Let’s say you are the lucky participant that gets selected to play the game out of the studio audience. The rules of the game are simple:
There are three doors. Behind two doors is a goat and behind one of the three doors is A BRAND NEW CAR! You get to take home either a goat or a car. (Why they used a goat rather than, say, a duck, is beyond me. Goats are just funnier I guess.)
You choose one of the three doors and proudly proclaim your choice of "Door Number One", "Door Number Two", or "Door Number Three".
After you’ve made your selection, Monty Hall would open one of the OTHER two other doors to reveal a goat. (out of the two remaining doors, one of them has to have a goat – possibly both of them if you happen to choose the door with the car).
You are then asked if you would like to stay with your original selection, or change your choice to the other un-opened door. Think carefully.
You hem and haw for a few minutes while the audience quivers in anticipation. Quivers, I tell you. You can feel the pressure growing.
After you decide to either stay with your original choice, or switch to the other door, the two remaining doors are opened and you either win a goat or A BRAND NEW CAR!
Easy enough, eh?
So here’s the crux of the puzzle … should you switch to the new door or stay with your first choice? The answer will surprise you.
Just for illustration, let's say you originally chose door number three. Monty then opens door number one to reveal a goat, and you have a choice of staying with door number three or switching your choice to door number two.
Most intelligent people would say that it doesn’t matter. Door number one has been eliminated and you are left with doors two and three – each of them, you think, has an equal chance of having the car. A straight up 50/50 chance. And although almost everyone says it doesn’t matter, the vast majority of people will choose to stay with their original choice.
WRONG ANSWER! In reality, you should ALWAYS switch doors. Your original choice only has a 33.3% chance of having the car behind it and the other door has a 66.6% chance of having the car. That’s the truth, I say, and I’m not just a Half Baked Lunatic!
Your brain is rebelling at this point, I know. You just can’t accept that answer. It’s stupid. You have two doors left, each with a 50/50 chance of having a car. Right? Wrong!!!
When I first encountered this problem and was shown the reasoning behind the answer, my mind just could not accept it. It took me about two days to really convince myself that yes – you should always switch when given the choice.
The problem was first popularized by Marilyn vos Savant, the high IQ columnist in Parade Magazine. She received over 10,000 letters saying she was wrong (admittedly, she didn't explain the answer very well - it was probably obvious to her but definitely not to everyone else, and she had to do a follow-up column to defend her answer and explain it to everyone in very simple terms). This was also featured in the popular book “The Curious Incident of the Dog in the Night-time” by Mark Haddon.
There are quite a few ways to look at the problem to understand the correct solution. As I said above, there is a psychological component with the way that the scenario is presented which re-directs your brain from seeing the real issue.
Here’s one way to think of it: suppose you are given the choice of selecting either one door or two doors to find the car. You’d select two doors, of course. If you have a choice of selecting one door or two doors to find the car, then there is a 33.3% chance of the car being behind the one door and a 66.6% chance of the car being behind one of the two doors.
The trick is, you were originally “steered” into choosing one door instead of two.
When you “switch” your choice, you are really choosing the other two doors. But remember that there is a 100% chance of a goat being behind ONE of the other two doors – so the fact that in the game they show you that one of the doors has a goat is irrelevant. It’s just a distraction.
"Ah", you say – "but opening one door changes the odds because there are only two doors left". No, I will tell you with only the slightest grin on my face, it doesn’t!
Another way to think about it: Pretend that the door you originally choose is your “anti-choice” – meaning you chose it to eliminate that choice, and you really want the other two doors. So you tell them you want door number three, but in your mind you want the 66.6% chance that the car is behind either door number one or door number two. It should be obvious to you that if they open all three doors at once, there is a 33.3% chance of the car being behind door number three and a 66.6% chance of the car being behind either doors one or two. Those odds don’t change if they open the doors one at a time ... and it doesn’t matter what order they open the doors, so if they open door one which they know will reveal a goat, the combination of doors one and two still have a 66.6% chance of having a car.
If you want to see a true mathematical proof of the results, just search for "Monty Hall Bayesian Analysis" and you can go through it step by step. There are a couple of other good "layman" explanations I've run across that help illustrate the correct answer.
Now that your brain is aching, go take an aspirin. Just make sure you get it from bottle number … oh, never mind.