The Monty Hall problem

Question

I was watching the movie $21$ yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the wrong answer) which revolves around theoretical probability.

The question goes a little something like this (I'm paraphrasing, but the numbers are all exact):

You're on a game show, and you're given three doors. Behind one of the doors is a brand new car, behind the other two are donkeys. With each door you have a $1/3$ chance of winning. Which door would you pick?

The character picks A, as the odds are all equally in his favor.

The teacher then opens door C, revealing a donkey to be behind there, and asks him if he would like to change his choice. At this point he also explains that most people change their choices out of fear; paranoia; emotion and such.

The character does change his answer to B, but because (according to the movie), the odds are now in favor of door B with a $1/3$ chance of winning if door A is picked and $2/3$ if door B is picked.

What I don't understand is how removing the final door increases the odds of winning if door B is picked only. Surely the split should be 50/50 now, as removal of the final door tells you nothing about the first two?

I assume that I'm wrong; as I'd really like to think that they wouldn't make a movie that's so mathematically incorrect, but I just can't seem to understand why this is the case.

So, if anyone could tell me whether I'm right; or if not explain why, I would be extremely grateful.

Answer 1

This problem, known as the Monty Hall problem, is famous for being so bizarre and counter-intuitive. It is in fact best to switch doors, and this is not hard to prove either. In my opinion, the reason it seems so bizarre the first time one (including me) encounters it is that humans are simply bad at thinking about probability. What follows is essentially how I have justified switching doors to myself over the years.

At the start of the game, you are asked to pick a single door. There is a $1/3$ chance that you have picked correctly, and a $2/3$ chance that you are wrong. This does not change when one of the two doors you did not pick is opened. The second time is that you are choosing between whether your first guess was right (which has probability $1/3$) or wrong (probability $2/3$). Clearly it is more likely that your first guess was wrong, so you switch doors.

This didn't sit well with me when I first heard it. To me, it seemed that the situation of picking between two doors has a certain kind of symmetry-things are either behind one door or the other, with equal probability. Since this is not the case here, I was led to ask where the asymmetry comes from? What causes one door to be more likely to hold the prize than the other? The key is that the host knows which door has the prize, and opens a door that he knows does not have the prize behind it.

To clarify this, say you choose door $A$, and are then asked to choose between doors $A$ and $B$ (no doors have been opened yet). There is no advantage to switching in this situation. Say you are asked to choose between $A$ and $C$; again, there is no advantage in switching. However, what if you are asked to choose between a) the prize behind door $A$ and b) the better of the two prizes behind door $B$ and $C$. Clearly, in this case it is in your advantage to switch. But this is exactly the same problem as the one you've been confronted with! Why? Precisely because the host always opens (hence gets rid of) the door that you did not pick which has the worse prize behind it. This is what I mean when I say that the asymmetry in the situation comes from the knowledge of the host.

Answer 2

To understand why your odds increase by changing door, let us take an extreme example first. Say there are $10000$ doors. Behind one of them is a car and behind the rest are donkeys. Now, the odds of choosing a car is $1\over10000$ and the odds of choosing a donkey are $9999\over10000$. Say you pick a random door, which we call X for now. According to the rules of the game, the game show host now opens all the doors except for two, one of which contains the car. You now have the option to switch. Since the probability for not choosing the car initially was $9999\over10000$ it is very likely you didn't choose the car. So assuming now that door X is a goat and you switch you get the car. This means that as long as you pick the goat on your first try you will always get the car.

If we return to the original problem where there are only 3 doors, we see that the exact same logic applies. The probability that you choose a goat on your first try is $2\over3$, while choosing a car is $1\over3$. If you choose a goat on your first try and switch you will get a car and if you choose the car on your first try and switch, you will get a goat. Thus, the probability that you will get a car, if you switch is $2\over3$ (which is more than the initial $1\over3$).

Answer 3

It's simple: switching allows you to pick 2 out of the 3 doors. Choosing door number 1 and then always switching, is the equivalent of saying "door number 2 or door number 3, but NOT door number 1". When you look at it that way, you should see that you have a 2/3 chance of being right, and that the opening of a door simply confirms which door it must be if you are right.

Increase the number of doors and it should become even more obvious that saying "door 2 or 3 or 4 or 5 or ..., but not 1" is the right way to bet. You have an $1-1/x$ chance of being right, and a $1/x$ chance of being wrong.