I finally understand the Monty Hall Problem

I first heard of the Monty Hall problem when I was in sixth grade.

The problem goes as follows:

Here are three doors. This is a game show. Here's the point. Behind one of the doors is a machine gun. You are a member of the game.

You choose one of the three doors. The main is to open one of the doors you didn't choose, while considering what is actually behind each door, and discover that there is nothing behind it.

Now you have the choice to either forget your personal first choice or run to another door that is not yet closed.

Statistics tell you that if you move to another door, your chances of actually getting the car during that time are 2/3, and 1/3 if you stay with the first door you chose. The question is why.

Me

I didn't understand what the problem was at the time. Looking at each case individually, the problem clearly makes sense. What concepts underlie this problem?

I decided to dig deeper. I went to college and started studying statistics with a 3. 9 GPA. Now I'm 20 years old. I spend all my free time studying statistics and mathematics, so I have no social life at all. I still haven't figured out how the Monty Hall problem is solved. In any case, every doctor and educator I've spoken to has reported it this way, and I didn't need to.

One day, while playing a video game, I discovered what was going on. After years of persistent work, I finally got to the essence of the problem and decided to write my conclusions here so that you don't have to spend the money I spent to understand it.

The concept

The mathematical explanation is very simple, but a little difficult to understand. Once you choose a starting door, the probabilities are basically "fixed". The probability that the car is behind the original door is equal to 1/3, and the probability that the car is behind one of the other two doors is equal to 2/3. If the main door opens one of the other doors without a car, there is a 2/3 chance that it will be "passed" to the other door, making the probability that the car is behind that door 2/3.

The ideas of "blocking" and "probability shifting" don't seem intuitively logical, but they've been explained online and elsewhere.

My explanation is a little easier to understand (I think). Suppose you participated in the Monty Hall game show several times. If you do not switch at least, you will set a machine beyond the first door you selected every time (probability is 1/3). If you switch every time, you will always rely on the fact that there is a machine behind either the first two doors you did not choose. If the car is beyond one of the doors, the host opens another door and finds that there is no car beyond, so it automatically switches to the door with a car.

The concept of solution is from a series of actions in this game show.

Before the decision is made, the moderator opens the door and indicates that there is no car behind, and having the questioner choose one from the remaining door, the probability of each choice is 1/2. 。 The selection of each door is equal to the ratio of the car behind it, and the probability is 1/2.

However, in this task, the participants choose the first door before the host opens one remaining door and indicates that there is no car behind it. Therefore, choosing another door does not mean to set the fact that there is a car behind this door, but other doors (one of them is already open before indicating that there is no car. The fact that there is a car behind) is to be set. From here, the concept of "rock" and "probability movement" is born. If the car is behind another door, it should be in front of the switched door.

He has a statistical degree and has no idea what to do in the future. I'm going to stop studying.

Understanding the Monty Hall Problem

The problem of Monty Hall is a common statistical puzzle.

• There are three doors, and there are two goats and one car behind it.
• You choose one door (called door A). Of course, you want it to be a car.
• Monty Hall, the moderator of the game show, examines other doors (B and C) and opens the goat with a goat. (If both doors have goats, choose randomly).

The game is as follows: Do you stay on the door A (first guess) or go to the indifferent door? Is this worth this?

Surprisingly, the win or loss is not 50 minutes. If you change the door, you can win with a 2/3 probability!

Today, let's find out why a simple game causes such a embarrassment. In fact, this game is made up of the revision of the decision when new information appears.

Play the game

If there are two doors, you will tweet that the probability is 50 minutes. Let's play a game:

Think about playing 50 times with Pick and Hold Strategy. Select the door 1 (or 2, 3) and keep ringing. Click and click. Let's look at your winning rate. You can see that it is actually formed within 1/3.

This time, restart and play 20 times with the pic k-an d-swap settings. If you select a door and Monty points to the goat (gray door), it will switch to another door. Let's look at your winning rate. Is it more than 50 %? Is it close to 60 %? Is it close to 66 %?

Stay & hold "strategies may produce good results in a small number of tests (less than 20 times or within the range). If you have one coin, how many times will it take to convince yourself that it's fair? Probably, two tables will appear in front of you, and you will judge that everything is fraudulent. Play the game several times, make the story equal, and reduce noise.

Understanding Why Switching Works

This is a difficult (but persuasive) way to understand that switching really works. Let's introduce a simpler way:

If I choose a door and maintain, I can win with a 1/3 probability.

My first hypothesis is on e-third. There are three random options.

No matter what, if you stick to your first option, you can't increase your potential. Monty has the option of adding 50 doors, there are options to explode others, and there are also options for dancing in Voodoo. On e-third of the probability of the first option I chose. Another door should be a different probability, that is, 2/3.

This comment may make sense, but there is no explanation of why the opposite probability is "improved". (Some readers have personal comments in the comments. Let them remember if the option of "1/3 remaining 2/3 is replaced").

Understanding The Game Filter

Let's take a look at why the switch looks good when you remove the door. Instead of playing a normal game, imagine the same options yourself:

• First there are 100 doors.
• You choose one door.
• Monty looks at the other 99 doors, finds a goat, and opens all the doors except one door.

Do you leave the door (1/100) at your start, or do you choose another door from the 99 door? (Try this with a simulation game. Use 10 doors instead of 100).

Monty is improved by removing 98 goats from the 99 varieties set. At the end of that, you can get the top door that can be selected from 99.

Choose a random door (first guess) from 100 or the best door from 99? In other words, do you want one random opportunity, or do you want the best in 99 random opportunities?

We begin to understand why Monty's influence help us. It gives you a chance to choose either a random general election or a filter election. There is a filter more than anything else.

But ... but ... shouldn't the two options mean 50/50 probability?

Overcoming Our Misconceptions

Assuming that two options mean 50/50 is the biggest barrier.

Yes, these two options can be the same without knowing anything. If I chose two Japanese pitchers at random and asked, "Which is the most highly evaluated?" Choose the names that sound the coolest, and about half and half would be the best. You don't know anything about the situation.

Let's say that pitcher A is a newcomer who has never been tested, and pitcher B has won the Best Player Award for the last 10 consecutive years. In this case, will your assumption change? Of course, you choose pitcher B (almost certainly). Your ignorant friends will still say this situation is 50 minutes.

The more you know…

This is a general idea. The more you know, the better you can make.

In the case of Japanese baseball players, you know more than friends and have a better chance. Rookies can become the best league player, but that's the probability. The more you test the old standard many times, the lower the possibility of a rookie.

This is exactly what happens in a game called "100 doors". The first choice is a random door (1/100), and the second choice is a champion (aka MVP of the tournament) that defeated the other 98 doors. It is likely that the champion is better than your door.

Visualizing the probability cloud

Imagine the filtering process. At first, every door has equal opportunities. Imagine a light green cloud that is evenly distributed in all doors.

When Monty begins to remove the bad candidates (from the 99 you did not choose), he pushed out the clouds from the bad door to the good door on the side. And the remaining doors have brighter green clouds.

After all the filters are over, your original door (still light green cloud) and the core green "championship" shining with the 98 door probability.

There is something important here: Monty is not trying to improve your door!

He is not intentionally investigating your door or trying to get rid of the goat. Rather, it's just "pull out weeds" from the grass next door.

Generalizing the game

The general principle is to review the victory as soon as new information is entered. for example

• The Baygien filter is improved with more information about whether the message is spam or not. It is not a good idea to stick to the original training dataset.
• Evaluating dogmas. Without any evidence, the two dogmas are identical. By gathering supporting corroboration (and running more tests), you can grow between belief in dogma A or B.

These are joint cases, but the meaning is clear: having more information actually means you overestimate your personal choices. The fatal flaw of the Monty Hall phenomenon is that you believe the probabilities are similar, since it blames the other door, and then you don't actually provide Monty filtering.

Summary

The key factors to be aware of the Monty Hall riddle are:

• The two options are 50/50, when you don't understand anything about it.
• Monty helps you by "filtering" the bad options on the other side. This is the choice between a random hypothesis and the "champion door" that is considered the best on the other side.
• In general, the more information you have, the more you actually overestimate your personal choices.

The fatal flaw of the Monty Hall phenomenon is that you believe the probabilities are similar, since it blames the other door, and then you don't actually provide Monty filtering. But the challenge is not to understand this conundrum, but to understand how further influences and information affect past conclusions. Fun math.

Appendix

To correct your awareness, consider other scenarios:

Your friend prepares a guess.

When you choose a door, Monty puts up a goat, but he doesn't understand how Monty reasoned.

He sees two doors and chooses 1:50/50 odds! He doesn't understand why one door should be more than all the others (but you do).

The main mess is that we fundamentally believe the same as our friend, that we inflate (or don't realize) the influence Monty has.

Monty's Wandering

When Monty opens the goat, the attack comes. He closes the door, mixes up all the prizes, and covers your door.

Will the switch help?

No. Monty starts filtering, but doesn't finish it. For example, you have 3 random choices like in the beginning.

Multi-valued Monty

Other Posts In This Series

1. Monty assigns you 6 doors. You choose one, and Monty divides the other five doors into groups 2 and 3. After that, he eliminates the goat, leaving no doors in one group. Where do you actually cross?
2. If there were three groups at the beginning. Your initial hypothesis contains 1/6 (16%), and there are two groups, so there is a 2/6 = 33% chance that it is correct.
3. Easy access to probability and statistical concepts
4. Intuitive (and concise) explanation of Bayes' axioms
5. Understanding Bayes' axioms with coefficient support
6. Understanding Monty Hall's difficulties

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