The Monty Hall Problem

Brief Description

Imagine that you are in a TV-show and the game host gives you a chance to win the grand prize. He asks you to pick out one from the three doors. Behind one of the doors is the prize, choosing the wrong door will leave you empty handed. You make your choice. Now, and that is the essence of the classical Monty Hall problem, the game host opens one of the doors that does NOT contain the prize. The game host knows where the prize is and MUST open one of the non prize containing doors. Then he asks if you want to stick with your original choice, or perhaps want to choose again.

So, there are two unopened doors before you and the question is, should you choose again (choose the other door) or go with the first choice. The Monty Hall problem solution is that you SHOULD CHOOSE AGAIN. Why? There seems to be an 50/50 chance to win the prize and seems like there will be no difference to which door to choose. Wrong. It is not a 50/50 chance (for the record, it’s 2/3 if door is changed and 1/3 when it isn’t) – it would be wise to choose again! Read on:

The Monty Hall Problem Explained

Entropy

There are numerous explanations as to why the player should choose again in the Monty Hall game. We don’t attempt to describe the multitude of them, but to propose a fairly original solution based on the entropy concept (at least extremely thorough Wikipedia article about Monty Hall paradox  doesn’t mention entropy at all).

Entropy is actually a rigorous mathematical concept from the statistical thermodynamics but we use its popular meaning here: entropy is the measure of disorder in the system. More entropy means a more disordered state. Untidy bathroom, for example, has bigger entropy content than the same bathroom when it’s clean and shiny. Low entropy situations are preferred to us – tidy bathroom, working car, beautiful wife etc.

Low entropy preferred outcome in the Monty Hall problem game would be to win the prize. The player of the Monty game doesn’t know which door hides the prize, but the game host knows. So, if the game host asks the player to reconsider the door choice, the player should do so. From the player’s point of view, the entropy of the Monty Hall conundrum is at its greatest at the beginning of the game as the player has no clue where the prize would be. But the game host knows where it is, and he can lessen the entropy of the Monty Hall game for the player.

When the game host opens a no prize door he decreases entropy (introduces negentropy) of the Monty Hall game state. For the player it means, that the possibility to win the prize just brightened up a bit – the game situation got better and player should choose the door again. Although the two unopened doors before the player look similar, one of them was chosen at the time the entropy was higher, and other door is a possibility arisen from the new situation with less inherent entropy.

Basically, the two remaining doors represent two different universes and the player should make the choice again (choose another door) in the “new and improved” universe. The game host has more information over the Monty Hall game than the player (player has none, because he has no clue where the prize could be) and due to the way he has to play the game, his actions are bound to decrease the entropy seen from player’s perspective and therefore improve the situation for the player. To better the winning chances in a Monty Hall game one should make one’s choice in the improved situation the game host has created by eliminating a door from the game.

The Monty Hall problem - all doors closed
At the beginning of the Monty Hall game all doors are closed and the player has 1/3 chance to win the prize
The Monty Hall problem - one door opened
Game host has opened a door that does not contain the prize. By doing so he reduces the entropy of the system and player should take advantage of that. Notice that the probability to win the prize has not changed for the player’s original door choice (middle cell) and still is 1/3 . The door the game host left for the player to reconsider has higher win chance (2/3).
The Monty Hall problem - incorrect perception of 50% win chance
Game host has opened a door but the player incorrectly perceives both remaining doors to have 50% win chance. Doors look similar, but the player ignores the circumstances that causes those two doors to be left in the game: one door represents player’s completely random guess while the other door represents game host’s informed action.
The Monty Hall problem - game end
Game is solved and placement of the prize becomes known to the player regardless of the win or not.

Metaphysical Musings

Multiple universes

At the heart of the Monty Hall problem is the question if one should change his initial choice one made at the start of the Monty Hall game. The stage of the Monty Hall game, where the game host asks the player to reconsider his initial choice is a strange state, where different outcomes of the game form a superposition. Basically three different universes meet and are tangled together. One universe, where the player made his initial choice (first image), one where the game host has opened a door (second image), and one, where the player changes his initial choice (if he chooses to do so). The three realities in three different universes will be at the superposition until the player forces one of the realities to be realized and the game is solved.

Examine the images above. They represent three different states of the Monty Hall problem game. First image describes the state before the game host has opened a no prize door. Second image is the state where the game host has eliminated a wrong door choice and reduced the entropy of the system. Last image represents the result of the game – either the player won the prize or not – its placement is now known to the player and the system reached its minimal entropy state seen from the player’s point of view (while nothing has changed for the game host as he always knew where the prize will be).

Things Can Look The Same But Never Are

Well, there you go, state of things is a matter of perspective. In our mind, the game host and the player are on the same level in the micro universe the Monty Hall game represents. But it is not so. Game host knows all about the game while the player knows nothing about it (at the beginning at least). We tend to abstract things and events to be same for the purpose of simplifying them. That’s why there is the big confusion understanding the Monty Hall problem solution. When the game host ask the player to reconsider, he wont see the direct need for it, because the two doors left in the game seem to be equal and if they are equal, why choose again?

But things are never equal, they change – in the Monty Hall door problem the two doors mark the different points in time with different entropy content. The state of the things is a matter of perspective, and the player’s perspective on the Monty Hall problem game changes with each action the player and game host are commencing. Time and state of the game both change by these actions (as the passing of time is the passing of events, it is correct to say, that time changes due to the game host and player actions in the Monty Hall paradox). The universe where the player makes his second and third choice in the Monty Hall game, is not the same universe, where he made the first choice. Things change.

For example, if you buy a car in 2020 is not the same thing as when you bought one in 1994. We abstract those two events to be similar just because they both are done in similar manner. Well, there IS a difference. Saturday and Tuesday, for example, seem to be just some days in the week, but in reality, they are unique events in the universe. And unique events have unique properties. Things that will happen on this Saturday are not the same as events that will take place on that Tuesday.

Buying a car in 2020 is not the same as buying one in 1994, because the economy has changed. And with that, the build quality of the cars has changed. It may be good in 2020 but it may have been better in 1994. The point in relation to the Monty Hall problem is that 1994 and 2020 are not equal when considering buying cars (they could be fairy equal in some other aspects tough). You should buy a car at times when the economy is at its high and the car manufacturers will (in general) make better cars, not at times where the economy is at its low and the cars are probably built lower quality. You can buy a good car in a low economy/low quality cars situation put your chances to do so are worse at times like this.

I guess the moral of the Monty Hall paradox game is that you should make your important decisions and take decisive action at times, when the things are looking good and prospects of success are higher. Right things at the right time, as they say. So, when you are in your “game of life” (like our imaginary player in the Monty Hall game) then recognize when the game host is dealing you a good hand, and don’t just stand there thinking, that all similar looking choices are equal in their essence, because they ain’t.