The Problem

I wonder if you have heard of this problem and the controversy it has caused.

Be very careful though as the problem appears to be simple at first but there is camouflaged within a nasty logic trap which has caught many people over the years.

The Problem is as follows:

You are on a game show and have just been shown three closed doors. The game show host tells you that there is a car hidden behind one of the doors and if you choose that door, you will win the car. The host also tells you that behind the other doors there are hidden goats, one goat per door. He tells you to choose a door and once you have, he opens one of the other doors to reveal a goat. There are now only two doors left and the host asks if you want to switch your selection to the other door. After you decide what you will do, you get to open your chosen door and see if you are a winner (this is my version of the problem as I wish you to have a fresh understanding of it, especially if you have seen it before).

Note: The host will always open a door and will always reveal one of the two goats. This is because he knows what is behind all the doors.

The Controversy

This problem was based on a United States game show called Let's Make a Deal and is named after the show's host Monty Hall.

It gained noteriety in 1990 when Marilyn vos Savant published an article about it in her Ask Marilyn Column in Parade.

Almost immediately she was flooded with strongly worded responses vigorously objecting to her stated answer. Many of these claiming to be professional mathematicians, teachers and university graduates.

Marilyn reported that initially only eight percent of the responses where in her favour and by the time she decided to stop discussing the problem this had only risen to fifty eight percent among those readers who had not performed any form of experiment to test the problem.

By then Marilyn had tried to show why she was correct by providing three separate descriptions of the solution.

Marilyn said the following: "Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance."

The majority of those unhappy readers where convinced that the probability of winning, after the door was opened to show a goat, was fifty percent, the same probability of getting a head, if you toss a coin once. So they said that Marilyn must be an idiot if she thought there was any advantage to switching doors.

At this point I wonder if you think much the same and that she must be wrong as well?

Many people I have explained this to have given me a hard time over it. And to be honest, until recently, I had trouble understanding it as well. Even Marilyn didn't explain it in any way that allowed me to see why she was correct. Since I first encounter this problem several years ago, I have seen a number of attempts by others, who also try to explain this problem, but none of them have succeeded in enlightening me.

Interestingly, of all the readers that had experimented to test the problem, ninety seven percent of them now agreed with Marilyn.

You can see Marilyn's article, including those nasty answers, on Marilyn's own website here: Game Show Problem

The Monty Hall Problem Explained

As I have shown above, I have been battling to understand the solution to the Monty Hall Problem for at least two years but after seeing the startling results for myself, via the iPhone App Monty Hall Paradox (MHP), I decided to try and solve it for myself, once and for all. Now I am going to share my insights with you.

Note: There are two Apps on the AppStore that will allow you to experiment with this problem. MHP is now ad supported and free but unfortunately it was just updated to version 1.1 and appears to be broken. I will post an article when it has been fixed, in the meantime I hope my explanation below puts your mind at ease. The other App is not free so I can not recommend it (unless you really want to try this for yourself, if so do a search in the AppStore for Monty Hall).

Anyway, I finally came up with a satisfactory explanation and a solution that I am happy to say, turns out to be logical, fairly simple and requires no special mathematically knowledge to understand. There is no magic here.

The Problem Again

I have rephrased the problem so that it is more formally stated:

• On a game show, a contestant is shown three doors and told that behind one of the doors there is a car and behind the other two doors there are goats
• He is told to win he must choose the door with the car
• He is now told to select a door, which he does and his selection is noted
• The host opens one of the two remaining doors and revels a goat
• The contestant is told he is must now open either one of the remaining doors to see if he has won

Question: Is there any advantage to switching and opening the door he did not first choose?

The Understandable Solution

First consider what happens when the initial selection is made by the contestant:

• The probability of the contestant selecting the door hiding the car is one out of three, or one third
• The probability of the contestant selecting a door hiding a goat is two out of three, or two thirds

Now consider what happens when the game show host opens one of the remaining doors that hides a goat:

• Two doors remain unopened
• One door hides the car
• One door hides a goat

Now consider what happens if the contestant remains with his first choice, that is declines to switch:

• If he has chosen the car, he wins, with a probability of one third
• If he has chosen a goat, he loses, with a probability two thirds

Now consider what happens if the contestant switches and chooses the other door:

• If he had chosen the car, probability one third, he will now get a goat and lose, with the same probability of one third
• If he had chosen a goat, probability two thirds, he will now get the car and win, with the same probability of two thirds

Note: It is important to understand that when the host opens the door with a goat, there is no change the original probability of selecting the car. There nothing special happening here, no magic.

QED (What does that mean anyway?)

Ok Why?

The key to understanding why this is the case, is to realise that switching does not cause the contestant to make a new random choice between two unopened doors (which would make the probability of winning the car one half, as many people believe) but it actually causes the contestant to, in effect, swap from one hidden object to the other.

In other words, if the car is behind the first door he selected, he will end up with a goat and lose, but if there was a goat behind that door he will win the car.

Put simpler, if he switches the doors, he switches the probability of selecting the car with that of selecting a goat (one third becomes two thirds).

So that is why the answer is yes and why there is an advantage to switching in this game.

I hope that this makes it clear for you. If you can understand this problem you can amaze and / or infuriate your family, friends and colleagues.

So Why is this So Hard to Get?

The following are just my assumptions but they are based my experience in both trying to understand this problem and from the reactions and arguments of others.

I believe that the paradox comes from a powerful misunderstanding that, once the door is opened, the solution is equal to that of tossing a coin and further, that this misunderstanding is combined with the intuitive and correct belief that opening the door can have no effect on the probability of the original selection. Though these two beliefs appear to back each other up, they are in fact mutually exclusive, as they contradict each other.

This misunderstanding also appears to be enhanced in people who are well versed in the principles of probability. I believe these people correctly see that there can be no real effect to the underlying probabilities, just because the door has been opened, but then find comfort in thinking that an even chance, of selecting the car, now that only two doors remain, demonstrates that there is no advantage to switching. If there was an advantage to switching, it follows that the probability must change and this is just not possible.

They just don't see they are stuck in a logic trap that will flip back and forth for ever. First they argue that probability does change, from one third to one half, when the door is opened, but then use that new probability to argue that the probability does not changed because the new one half probability demonstrates there is no advantage to switching.

All this reminds me of the logic trap that is suppose to be able to crash an intelligent computer: "If I am a lawyer and all lawyers are liars, am I telling the truth?".

I wonder what the intelligent computer would make of the Monty Hall Problem?

Safe Surfing!