What is the show 3 doors. Paradox Monty Hall - Logic problem not for weaklikov

About lotteries

This has long acquired a massive character and has become an integral part of modern life. And although the lottery is increasingly expanding its capabilities, many people still see it only a way to enrich. Let not free and not reliable. On the other hand, as one of the heroes of Jack London noted, it is impossible not to reckon with the facts in the gambling, people are sometimes lucky.

Mathematics case. History of probability theory

Alexander Buffetov

Transcript and video recording of the lecture of the doctor of Physical and Mathematical Sciences, the leading researcher of the Mathematical Institute named after Steklov, the leading scientific officer of the IPI RAS, Professor of the Faculty of Mathematics of the Higher School of Economics, Director of the National Center for Scientific Research in France (CNRS) Alexander Buffetova, read as part of the cycle " Public lectures "Polit.ru" "February 6, 2014

Illusion of regularity: why the accident seems unnatural

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Determinism

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God is statistics

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What is determinism

If the initial system conditions are known, it is possible using the laws of nature, to predict its final state.

The task of the tearing bride

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The dependence of the growth of the child from parental growth

Young parents, of course, want to know what growth will their child, becoming adults. Mathematical statistics may offer a simple linear dependence for an approximate growth assessment of children, based on the growth of father and mother, as well as indicate the accuracy of such an assessment.

Paradox Monty Hall - probably the most famous paradox in probability theory. There is a mass of its variations, for example, the paradox of three prisoners. And there is a lot of interpretations and explanations of this paradox. But here, I would like to give not only a formal explanation, but to show the "physical" basis of what is happening in the Paradox of Monti Hall and the like.

Classical wording is as follows:

"You are a participant in the game. Before you three doors. For one of them a prize. The presenter invites you to try to guess where the prize. You specify one of the doors (at random).

Paradox wording Monti Hall

The presenter knows where the prize is actually. He, until, does not open the door to which you have shown. But it opens up one of the remaining doors, behind which there is no prize. The question is whether you should change your choice, or stay with the previous decision? "

It turns out that if you simply change the choice, then your chances will be won old!

The paradoxicity of the situation is obvious. It seems that everything is happening by chance. There is no difference, you will change your decision or not. But it is not.

"Physical" explanation of the nature of this paradox

Let's first, we will not go into mathematical subtleties, but simply not biased to look at the situation.

In this game you only first make a random choice. Then the host informs you additional informationwhich allows you to increase your chances of winning.

How does the presenter tell you additional information? Very simple. Note that it opens not any a door.

Let us, for simplicity (at least there is an element of the lucavia), consider a more likely situation: you have shown on the door behind which there is no prize. Then, for one of the remaining doors a prize there is. That is, the leading has no choice. He opens a completely definite door. (You specified for one, for the other there is a prize, there remains only one door that the presenter can open.)

It is at this moment of meaningful choice, he informs you the information you can take advantage.

In this case, the use of information is that you change the solution.

By the way, your second choice is also not accident (Rather, not for so much case as the first choice). After all, you choose from closed doors, and one is already open and she not arbitrary.

Actually, after these reasoning, you may have a feeling that it is better to change the decision. This is true. Let's show it more formally.

A more formal explanation of the Paradox of Monti Hall

In fact, your first, random, the choice splits all the doors into two groups. Behind the door that you chose a prize is with a probability of 1/3, for two others - with a probability of 2/3. Now the lead makes changes: it opens one door in the second group. And now the whole probability of 2/3 applies only to a closed door from a group of two doors.

It is clear that now you benefit from changing your decision.

Although, of course, you have a chance to lose.

Nevertheless, the selection of choice increases your chances of winning.

Paradox Monty Hall.

The Paradox of Monty Hall is a probabilistic task, the solution of which (according to some) contradicts common sense. Task wording:

Imagine that you have become a participant in the game in which you need to choose one of the three doors. For one of the doors there is a car, behind two other doors - goats.
You choose one of the doors, for example, number 1, after that a lead that knows where the car is located, and where - goats, opens one of the remaining doors, for example, number 3, followed by a goat.

Paradox Monty Hall. The most inaccurate mathematics

After that, he asks you if you do not want to change your choice and choose the door number 2.
Will your chances of winning the car will increase if you accept the offer offer and change your choice?

When solving the problem, it is often mistaken that two choices are independent and, therefore, the probability of changing the choice will not change. In fact, this is not the case, in what you can make sure I remember the Bayes formula or looking at the simulation results below:

Here: "Strategy 1" - not to change the choice, "Strategy 2" - change the choice. Theoretically, for the case with 3 doors, probabal distribution - 33, (3)% and 66, (6)%. With numerical simulation, we would have to get similar results.

Links

Paradox Monty Hall. - The task from the partition of the theory of probability, in which the contradiction of common sense is viewed.

History of the emergence [edit | edit wiki text]

At the end of 1963, a new current show called "Let's Make A DEAL" ("Let's agree"). According to the scenario, the audience from the audience received prizes for the right answers, having a chance to increase them, making new bets, but risking the wins. The founders of the show were Stephen Khatosu and Monti Hall, the last of which became his unchanged leading for many years.

One of the tasks for the participants was the drawing of the main prize, which was located at one of the three doors. In two remaining incentive prizes, in turn, the lead knew the order of their location. The participant was necessary to determine the winning door, putting his own winnings for the show.

When the guessing was determined with the number, the presenter opened one of the remaining doors, behind which there was an incentive prize, and offered a player to change the originally selected door.

Wording [edit | edit wiki text]

As a specific task, the paradox first formulated Steve Selvin (Steve Selvin) in 1975, which went to the magazine The American Statistician (American Statistics), and the leading Monti Hall, the question: whether the participant chances will change to win the main prize, if after opening the door with In encouragement, he will change his choice? After this incident, the concept of "Paradox Monti Hall" appeared.

In 1990, it was in Parade Magazine (Parade magazine) published the most common version of the paradox with an example:

"Imagine yourself on a teleigre, where you need to give preference to one of the three doors: for two of them goats, and for the third - a car. When you choose, assuming, for example, that the winning door number one one, the presenter opens one of the remaining two doors, for example, the number three, behind which the goat. Then give you a chance to change the choice on another door? Is it possible to increase the chances of winning the car if you change your choice from the door number one door number two? "

This formulation is a simplified option, because There is a factor of the influence of the lead, who knows exactly where the car is interested in losing the participant.

So that the task has become a purely mathematical, it is necessary to eliminate the human factor, by entering the opening of the door with an incentive prize and the ability to change the original choice as essential conditions.

Solution [edit | edit wiki text]

When comparing the chances at first glance, the change of the door number will not give any advantages, because All three options have a chance for winning 1/3 (approx. 33.33% per each of three doors). At the same time, the discovery of one of the doors will not affect the chances of the two remaining whose chances will be 1/2 to 1/2 (50% per two remaining doors). The basis of such a judgment is the judgment that the choice of door by the player and the choice of doors leads - two independent events that do not affect one thing. In fact, it is necessary to consider the entire sequence of events as a whole. In accordance with the theory of the probability, the first selected door chances from the beginning and until the end of the game are consistently 1/3 (ok.33.33%), and in the two remaining 1/3 + 1/3 \u003d 2/3 (approx. 66.66%). When one of the two remaining doors opens, its chances are getting 0% (an incentive prize is hidden behind it), and as a result, the chances of a closed unbalanced door will be 66.66%, i.e. Twice more than the originally selected.

To facilitate the understanding of the selection results, one can consider an alternative situation in which the number of options will be more, for example - a thousand. The probability of choosing a winning version will be 1/1000 (0.1%). Provided that later, nine hundreds of the remaining nine hundred and ninety-nine options will be discovered nine hundred and ninety-eight incorrect, it becomes obvious that the likelihood of one remaining door of nine hundred and ninety-nine unrefected higher than that of the only one chosen at the beginning.

Mention [edit | edit wiki text]

You can meet the mention of the Paradox of Monti Hall in the "Twenty One" (film Robert Luketich), "Notep" (Roman Sergey Lukyanenko), TV series "4Isla" (television series), "Mysterious Night Murder of Dog" (Tale of Mark Haddon), "XKCD" ( Comic), "Destroyers of Legends" (TV show).

See also [edit | edit wiki text]

In the image, the process of choice between two buried doors of the three proposed originally

Examples of solutions for problems on combinatorics

Combinatorics - This is a science, with which everyone meets in everyday life: how many ways to choose 3 attendant to clean the class or how many ways to make a word from these letters.

In general, the combinatorics allows you to calculate how many different combinations, according to some conditions, can be made from the specified objects (identical or different).

As a science, the combinatorics arose in the 16th century, and now every student studies it (and often even a schoolboy). Studying from the concepts of permutations, accommodations, combinations (with or without repetitions), you will find tasks for these topics below. The most famous rules of combinatorics are the rules of the amount and works that are most often used in typical combinatorial tasks.

Below you will find several examples of tasks with solutions for combinatorial concepts and rules that will make it possible to deal with typical tasks. If there are difficulties with tasks - order the control according to the combinatorics.

Combinatorics tasks with online solutions

Task 1. Mom has 2 apples and 3 pears. Every day, for 5 days in a row, she gives one fruit every day. How many ways can it be done?

Solution of the challenge of combinatorics 1 (PDF, 35 KB)

Task 2. The company can provide work for one specialty 4 women, on the other - 6 men, on the third - 3 employees regardless of gender. How many ways can be fill in vacant places if there are 14 applicants: 6 women and 8 men?

Solution of the Combinatorics Task 2 (PDF, 39 KB)

Task 3. In the passenger train of 9 cars. How many ways can be searched in a train of 4 people, provided that they all have to go to various cars?

Solution of the problem of combinatorics 3 (PDF, 33 KB)

Task 4. In a group of 9 people. How much can the form of different subgroups, provided that at least 2 people are included in the subgroup?

Solution of the Combinatorics Task 4 (PDF, 34 KB)

Task 5. A group of 20 students need to be divided into 3 brigades, and 3 people should be included in the first brigade, in the second - 5 and third - 12. In how many ways it can be done.

Solution of the problem of combinatorics 5 (PDF, 37 KB)

Task 6. To participate in the team, the coach selects 5 boys out of 10. how many ways can it form a team if 2 certain boys must enter the team?

Task on combinatorics with decision 6 (PDF, 33 KB)

Task 7. 15 chess players took part in the chess tournament, and each of them played only one batch with each other. How many parties were played in this tournament?

Task by combinatorics with decision 7 (PDF, 37 KB)

Task 8. How many different fractions can be made up from numbers 3, 5, 7, 11, 13, 17 so that 2 different numbers come into every fraction? How many of them will be the right fractions?

Combinatorics task with decision 8 (PDF, 32 KB)

Task 9. How many words can I get, rearranged letters in the word Mountain and Institute?

Task on combinatorics with decision 9 (PDF, 32 KB)

Task 10. What numbers from 1 to 1 000 000 more: those in which a unit occurs, or those in which it does not occur?

Task on combinatorics with decision 10 (PDF, 39 KB)

Ready examples

Need solved problems on combinatorics? Find in Reshebnik:

Other solutions to the tasks on probability theory

Imagine that a banker invites you to choose one of three closed boxes. In one of them 50 cents, to another - one dollar, in the third - 10 thousand dollars. What choose, that you will come as a prize.

You choose at random, say, box number 1. And then the banker (who naturally knows where what) right on your eyes opens a box with one dollar (for example, it's No. 2), after which it invites you to change the originally selected box # 1 on the box number 3.

Should you change your decision? Will your chances of getting 10 thousand?

This is the paradox of Monty Hall - the task of the theory of probability, the solution of which, at first glance, contradicts common sense. Through this task, people break their heads since 1975.

The paradox was called in honor of the leading popular American television show "Let's Make A Deal". In this TV show there were similar rules, only participants chose the doors, for two of which goats were hiding, for the third - Cadillac.

Most players argued that after the closed doors were left two and one of them is Cadillac, the chances of getting it 50-50. Obviously, that when the lead opens one door and offers you to change your decision, he begins a new game. You will change the solution or do not change, your chances will still be 50 percent. So?

It turns out that there is no. In fact, changing the decision, you double the chances of success. Why?

The most simple explanation of this answer consists in the following consideration. In order to win the car without changing the choice, the player must immediately guess the door behind which the car stands. The probability of this is 1/3. If the player originally falls on the door behind which the goat is (and the probability of this event is 2/3, since there are two goats and only one car), then he can definitely win the car by changing his solution, as the car and one goat remain, And the door with the goat was already opened.

Thus, without changing the choice, the player remains with its initial probability of winning 1/3, and when the initial choice change, the player wores himself two times the remaining likely that at the beginning he did not guessed.

Also, an intuitive explanation can be made by changing two events in places. The first event is to make a decision by the player about the door change, the second event is the opening of an excess door. This is permissible, since the opening of an excess door does not give the player any new information (cm in this article). Then the task can be reduced to the following wording. At first point, the player divides the door into two groups: in the first group one door (the one he chose), in the second group the two remaining doors. The following time, the player makes a choice between groups. Obviously, for the first group, the probability of winning 1/3, for the second group 2/3. The player chooses the second group. In the second group, he can open both doors. One opens the lead, and the second player himself.

Let's try to give "the most understandable" explanation. We reformulate the task: the honest lead announces the player that behind one of the three doors is a car, and it offers it first to point out one of the doors, and then choose one of two actions: open the specified door (in the old wording it is called "not to change your choice ") Or open the other two (in the old wording it will be" to change the choice ". Think, here is the key to understanding!). It is clear that the player will choose a second of two actions, since the probability of obtaining a car in this case is twice as high. And the one thing that leading before the selection of the action "showed the goat," does not help and does not interfere with the choice, because at one of the two doors there is always a goat and the host will surely show it at any course of the game, so the player can on this goat and Do not watch. The player's case, if he chose the second action - say "Thank you" to the lead that he saved him to open one of the two doors himself, and open another. Well, or even easier. Imagine this situation from the leading point of view, which makes such a procedure with dozens of players. As he knows perfectly well what is behind the doors, then, on average, in two cases out of three, he sees in advance that the player chose the door "not that". Therefore, for him, there is definitely no paradox in the fact that the correct strategy is to change the choice after the opening of the first door: then in the same two cases out of three player will leave the studio on the new car.

Finally, the most "naive" proof. Let the one who stands on his choice is called "stubborn", and the one who follows the instructions of the leader is called "attentive." Then the stubborn wins if he initially guessed the car (1/3), and attentive - if he was at first missed and hit the goat (2/3). After all, only in this case, then he will point on the door with the car.

Monti Hall, Producer and Lead Show Let's make a deal From 1963 to 1991.

In 1990, this task and its decision was published in the American magazine "Parade". The publication caused a flurry of indignant reviews of readers, many of which possessed scientific degrees.

The main complaint was that not all the terms of the task were stipulated, and any nuance could affect the result. For example, the presenter could suggest changing the decision only if the player first chose a car. Obviously, the change of initial choice in such a situation will lead to a guaranteed loss.

However, since the very launch of the television show, Monti Hall people who changed the solution really won twice as often:

Of the 30 players who have changed the initial decision, Cadillac won 18 - that is, 60%

Of the 30 players who stayed with their choice, Cadillac won 11 - that is, about 36%

So given in the decision of the reasoning, no matter how illogical they seemed to be confirmed by practice.

Increase the number of doors

In order to make it easier to understand the essence of what is happening, you can consider the case when the player sees no three doors in front of him, but, for example, a hundred. At the same time, there is a car for one of the doors, and for the rest of 99 - goats. The player chooses one of the doors, while in 99% of cases he will choose the door with the goat, and the chances immediately choose the door with the car are very small - they make up 1%. After that, the presenter opens 98 doors with goats and offers the player to choose the remaining door. At the same time, in 99% of cases, the car will be behind this remaining door, since the chances of the fact that the player immediately chose the right door, very small. It is clear that in this situation a rationally thinking player should always accept the proposal of the lead.

When considering an increased number of doors, the question often arises: if the leader opens one door of three in the original task (i.e. 1/3 of the total number of doors), then why it is necessary to assume that in the case of 100 doors, the presenter will open 98 doors with goats, and not 33? This consideration is usually one of the significant reasons why the Paradox of Monty Hall is contrary to the intuitive perception of the situation. Explain the opening of 98 doors will be correct because a significant condition of the task is the presence of only one alternative choice for a player who is proposed by the lead. Therefore, in order for the tasks similar to, in the case of 4 doors, the presenter should open 2 doors, in the case of 5 doors - 3, and so on, so that one unopened door always remains except that the player originally chose. If the presenter will open a smaller number of doors, then the task will no longer be similar to the original Monti Hall task.

It should be noted that in the case of a set of doors, even if the presenter will leave a closed not one door, but a few, and offer the player to choose one of them, then when changing the initial choice, the player's chances win the car will still increase, although not so much. For example, consider the situation when the player chooses one door from the hundred, and then the presenter opens only one door from the remaining, offering the player to change its choice. At the same time, the chances of the fact that the car is at the originally chosen player by the door, remain the same - 1/100, and for the rest of the doors, the chances change: the total probability that the car is behind one of the remaining doors (99/100) is now distributed now 99 doors, and on 98. Therefore, the probability of finding a car for each of these doors will be equal to 1/100, but 99/9800. The incidence of probability will be approximately 1%.

The Tree of Possible Player Solutions and the Master, showing the likelihood of each outcome. More formally, the game scenario can be described with the help of decisions. In the first two cases, when the player first chose the door behind which the goat is, the change in the choice leads to the win. In the last two cases, when a player first chose a door with a car, a change in the choice leads to the loss.

If you do not understand anyway, spit on formulas and justcheck out everything statistically. Another explanation option:

• A player whose strategy would be to change the chosen door every time, will only lose if he originally chooses the door behind which the car is located.
• Since the probability of choosing a car on the first attempt is one to three (or 33%), then the chance does not choose a car if the player has changed its choice, is also one to three (or 33%).
• This means that the player who used the strategy to change the door will won with a probability of 66% or two to three.
• It will double the chances of winning the player whose strategy - every time you do not change your choice.

Still do not believe? Suppose you chose the door number 1. Here are all possible options for what can happen in this case.

We all familiarize the situation when we instead of a sober calculation relied on our intuition. After all, you need to admit that it is not always possible to calculate everything before making a choice. And no matter how Lukvali people who are accustomed to making their choice only after careful analysis, none once had to do it according to the principle "probably so." One of the reasons for this action may be a banal absence of the required time to assess the situation.

At the same time, the choice is waiting for the current situation right now, and does not allow to get away from the answer or action. But even more trusted situations for us, which literally causes the convulsion of the brain, is the destruction of confidence in the correctness of the choice or in its likely superiority over other options based on logical conclusions. All existing paradoxes are based on it.

Paradox in the game Teleow "Let's Make A Deal"

One of the paradoxes, which causes hot disputes among puzzle lovers, is called the Paradox of Monti Hall. He is named after the leading television show in the US called "Let's Make A Deal". At the TV show, the host proposes to open one of the three doors, where the car is located as a prize, while at the other two are on the same goat.

The participant of the game makes his choice, but leading, knowing where the car is located, does not open the door that the player pointed out, and the other in which the goat is located and offers to change the original choice of the player. For further disclaim, we accept this particular version of the leading behavior, although in fact it can periodically change. Other development scenarios We will simply list below in the article.

What is the essence of the paradox?

Once again, on points, we denote the conditions and change the objects of the game for a variety on your own.

Member of the game are indoors with three bank cells. In one of the three cells, the gold ingot of gold, in the other two one coin with a par with 1 kopeck of the USSR.

So, the participant before choosing and the conditions of the game is as follows:

1. The participant can choose only one of three cells.
2. The banker knows the initially the location of the ingot.
3. The banker always opens a cell with a coin that is different from the player's choice, and suggests changing the choice of a player.
4. The player may, in turn, change its choice or leave the original.

What does the intuition say?

The paradox is that for most people who are used to thinking logically, the chances of winning in case of a change of their initial choice 50 to 50. After all, after the banker opens another cell with a coin, different from the initial choice of the player, 2 cells remain in one of which is the ingot of gold, and in another coin. The player wins the ingot if the banker's offer accepts the cell under the condition if there was no ingot in the initially chosen player in the cell. And on the contrary, with this condition, it loses, if it refuses to accept the offer.

As we suggest common sense, the likelihood of choosing a ingot and winnings in this case is 1/2. But actually the situation is different! "But how so, is it all obvious?" - you ask. Suppose you chose a cell number 1. Intuitively, it does not matter what choice did you have originally, in the end, you have in fact before choosing a coin and ingot. And if initially you had the likelihood of obtaining the prize 1/3, then ultimately, when opening one cell, the banker you get a probability 1/2. It seemed probability increased from 1/3 to 1/2. With a careful analysis of the game, it turns out that when changing the solution, the probability increases to 2/3 instead of intuitive 1/2. Let's look at what happens.

Unlike an intuitive level, where our consciousness considers the event after changing the cell as something separate and forgets about the initial choice, mathematics does not break these two events, but on the contrary, retains the chain of events from start to the end. So, as we have previously been talking, the chances of winning when you get to the ingot from our 1/3, and the likelihood that we will choose a cell with a 2/3 coin (since we have one ingot and two coins).

1. We choose the initially bank cell with a fusion - the probability of 1/3.
   • If the player changes his choice, taking the banker's offer, he loses.
   • If the player does not change the choice, without taking the banker's offer, he wins.
2. We choose from the first time a bank cell with a coin - probability 2/3.
   • If a player has changed his choice - won.
   • If the player does not change the choice - lost.

So, in order for the player to leave the bank with a gold stylet in his pocket, he must choose a remotely losing position with a coin (probability 1/3), and then accept the banker's offer to change the cell.

In order to understand this paradox and escape from the shackles of the initial selection template and the remaining cells, let's imagine the player's behavior by an even account on the contrary. Before the banker suggests a cell to choose, the player mentally defined precisely with the fact that he changes his choice, and only after that it should be an event opening an excess door. Why not? After all, the open door does not give for him more information in such a logical sequence. At the first stage of time, the player shares the cells into two different areas: the first is a domain with one cell with its original choice, the second with two remaining cells. Next, the player has to make a choice between two regions. The probability of getting from the cell a gold ingot from the first area of \u200b\u200b1/3, from the second 2/3. The choice follows the second area in which it can open two cells, the banker will open the first one, he himself.

There is an even more understandable explanation of the Paradox of Monti Hall. To do this, it is necessary to change the wording of the task. The banker makes it clear that in one of the three banking cells there is a gold ingot. In the first case, he offers to open one of the three cells, and in the second - at the same time two. What will the player choose? Well, of course two at once, by increasing the probability doubled. And the moment when the banker opened a cell with a coin, this player actually does not help and does not interfere with the choice, because the banker will show this cell with a coin anyway, so the player can simply ignore this action. On the player, you can only thank the banker for facilitating his life, and instead of two he had to open one cell. Well, finally, you can get rid of the paradox syndrome if you put yourself in the place of the banker, which initially knows that a player in two of three cases indicates the wrong door. For the banker, the paradox is absent as such, because it is sure in such an inversion of events that the player takes the golden stumps in case of changing events.

The Paradox of Monty Hall clearly does not allow to be in winning conservatives that are reinforcedon with its original choice and lose their chance of growth likely. For the conservatives, it will remain 1/3. For vigilant and reasonable people, he grows up to the above 2/3.

All these statements are relevant only in compliance with the initially stipulated conditions.

What if you increase the number of cells?

What if you increase the number of cells? Suppose instead of three there will be 50. The gold ingot will lie only in one cell, and in the remaining 49 - coins. Accordingly, in contrast to the classical case, the likelihood of hitting the goal 1/50 or 2% instead of 1/3, while the probability of choosing a cell with a coin is 98%. Next, the situation is developing, as in the same case. The banker offers to open any of the 50 cells, the participant chooses. Suppose the player opens the cell under the sequence numbers 49. The banker in turn, as in the classical version, is not in a hurry to fulfill the desire of the player and opens the other 48 cells with coins and offers to change their choice for the remaining 50.

It is important to understand that the banker opens up 48 cells, and not 30, and leaves 2, including the chosen by the player. It is this choice that allows a paradox to go into incision with intuition. As in the case of a classic option, the opening of a banker 48 cells leaves only one single alternative for choice. The case of an option of a smaller opening of the cells does not allow you to put a task with classics in one row and feel a paradox.

But since we also touched this option, let's assume that the banker leaves not one other than the chosen player, but several cells. Presented, as before, 50 cells. The banker after choosing a player opens only one cell, while leaving 48 cells closed, including the chosen by the player. The probability of choosing a ingot from the first time is 1/50. In total, the probability of finding the ingot in the remaining 49/50 cells, which in turn is extracting not at 49, but by 48 cells. It is not difficult to calculate that the probability of finding the ingot in this embodiment is equal to (49/50) / 48 \u003d 49/2900. The probability is not much for a lot, but still higher than 1/50 by approximately 1%.

As we mentioned at the very beginning of the lead Monti Hall in the classic game scenario with doors, goats and a prize car can change the conditions of the game and with it and the probability of winning.

Mathematics Paradox

Can mathematical formulas prove an increase in likelihood when changing the choice?
Imagine the chain of events in the form of a set divided into two parts, the first part of the X is the selection at the first stage of the package of the safe player; And the second set Y is the remaining two other cells. The probability (c) of winning for cells 2 and 3 can be expressed using formulas.

In (2) \u003d 1/2 * 2/3 \u003d 1/3
In (3) \u003d 1/2 * 2/3 \u003d 1/3

Where 1/2 is the probability with which the banker will open cell 2 and 3, provided that the player initially selected the cell without a ingot.
Further, the conditional probability of 1/2 when the banker is opened with a coin cell varies by 1 and 0. Then the formulas acquire the following form:

In (2) \u003d 0 * 2/3 \u003d 0
B (3) \u003d 1 * 2/3 \u003d 1

Here we clearly see that the probability of choosing the ingot in the cell 3 - 2/3, and this is just over 60 percent.
The programmer of the initial level can easily verify this paradox by writing a program that considers the likelihood when changing the choice or vice versa and refer results.

Explanation of the paradox in the film 21 (twenty one)

The visual explanation of the Paradox of Monti Paul is given in the film "21" (twenty-one), director Robert Lucotich. Professor Mickey Dew at the lecture brings an example from the LET's Make A Deal show and asks a question about the probability of the probability of Student Ben Campbell (actor and singer James Anthony), which gives the right alignment and thus surprises the teacher.

Independent study of paradox

For people who want to check out the result independently, but not having a mathematical basis, we offer to simulate the game yourself in which you will lead, and someone will be a player. You can use in this game of children who will choose candy or candy from them in advance prepared cardboard boxes. Each choice, be sure to fix the result for further counting.

Ecology of knowledge. One of the tasks of probability theory is the most interesting and seemingly contrary to the common sense of the Paradox of Monty Hall, named so in honor of the leading American television show "Let's Make A Deal".

Many of us probably heard about the theory of probabilities - a special section of mathematics, which studies patterns in random phenomena, random events, as well as their properties. And just one of the tasks of probability theory is the most interesting and, it would seem, contrary to common sense, the Paradox of Monty Hall, named so in honor of the leading American television show "Let's Make A Deal". With this paradox we want to introduce you today.

Definition of Paradox Monte Hall

As the task of the Paradox of Monty Hall is defined as the descriptions of the above-mentioned game, the most common among which is the wording, which was published by the magazine Parade Magazine in 1990.

According to her, a person must introduce himself to the participant of the game where you need to choose one door of three.

There is a car behind one door, and for the rest - goats. The player must choose one door, for example, door number 1.

A leader who knows what is behind each door opens one of the two doors, which remained, for example, the door number 3, behind which the goat is.

After that, the lead is interested in the player, does he not want to change his original choice and choose the door number 2?

Question: Will the player chances rise if he change his choice?

But after the publication of this definition, it turned out that the task of the player was formulated somewhat incorrectly, because Not consistent all the conditions.

For example, the leading game can choose the "Hell Monti" strategy, offering to change the choice only if the player initially guess the door behind which the car is located.

And it becomes clear that the change in the choice will lead to one hundred percent loss.

Therefore, the greatest popularity was obtained by setting the problem with a special condition No. 6 from a special table:

• The car can be with the same probability to be behind each door.
• The lead is always obliged to open the door with the goat, except the player who chose, and offer the player the ability to change the choice
• Host, having the opportunity to open one of two doors, chooses anyone with the same probability

Presented below, the analysis of the Paradox of Monty Hall is considered precisely taking into account this condition. So, the analysis of the paradox.

Hall Paradox Paradox

There are three developments of events:

During the solution of the task presented, such arguments are usually given: the lead in each case removes one door with the goat, therefore, the probability of finding a car for one of the two closed doors is equal to ½, no matter what choice was made initially. However, it is not.

The meaning is that, making the first choice, the participant shares the doors to A (selected), B and C (remaining). Chances (P) on the fact that the car stands behind the door A is equal to 1/3, and on the fact that it is behind the doors B and C are equal to 2/3. And the chances of success when choosing doors B and C are calculated as follows:

P (b) \u003d 2/3 * ½ \u003d 1/3

P (C) \u003d 2/3 * ½ \u003d 1/3

Where ½ is a conditional probability that the car is behind this door, provided that the car is not behind that door that the player chose.

The presenter, opening a deliberately losing door of the two remaining, informs the player 1 bit of information and thus changes the conditional probabilities for the doors B and C on the values \u200b\u200bof 1 and 0. Now the chances of success will be calculated as follows:

P (B) \u003d 2/3 * 1 \u003d 2/3

P (C) \u003d 2/3 * 0 \u003d 0

And it turns out that if the player changes its original choice, its chance of success will be equal to 2/3.

This explains this as follows: By changing your choice after the leader's manipulations, the player will win if initially he chose the door with the goat, because The presenter opens the second door with the goat, and the player remains only to change the doors. You can select the door with the goat in two ways in two ways (2/3), respectively, if the player replaces the door, then wins with a probability of 2/3. It is because of the contradictions of this withdrawal with intuitive perception of the task and received the status of a paradox.

Intuitive perception speaks of the following: when the lead opens a losing door, a new challenge gets up in front of the player, at first glance, not related to the initial choice, because The goat for the opened drive door will be there anyway, regardless of whether the player or winning door initially chose a player.

After opening the master door, the player must make a choice again - either to stay on the former doors, or choose a new one. This means that the player does just a new choice, and does not change the original one. And the mathematical solution addresses two consecutive and related tasks of the master.

But you need to keep in mind that the presenter opens the door from those two that remained, but not the one that chose a player. So, the chance for the fact that the car is behind the remaining door increase, because Presenter did not choose it. If the lead knows that the goal behind the door chosen by the player is, it will still open it, it will also know how the player will choose the right door, because the probability of success becomes ½. But this is already a game for other rules.

And here's another explanation: Suppose the player plays according to the system presented above, i.e. From the doors B or C always chooses the one that differs from the initial selection. He will lose if it originally chose the door with the car, because Subsequently chooses the door with the goat. In any other case, the player will win if initially chose a losing option. However, the likelihood that initially he will choose it, is 2/3, from which it follows that for success in the game you first need to make a mistake, the likelihood of which is twice as much as the probability of the right choice.

Third explanation: Imagine that the doors are not 3, and 1000. After the player made a choice, the lead removes 998 unnecessary doors - only two doors remain: chosen by the player and one more. But the chance for the fact that the car for each of the doors is not at all ½. Most likely (0.999%) the car will be behind that door that the player did not choose initially, i.e. Behind the door selected from the remaining after the first choice of 999 others. Approximately needed and arguing when choosing from three doors, let the chances of success and decline and become 2/3.

And the last explanation is the replacement of conditions. Suppose that instead of doing the original choice, for example, doors number 1, and instead of opening the door number 2 or number 3, the player must make a correct choice from the first time, if he knows that the probability of success with the door number 1 is equal to 33 %, but about the absence of a car outside the door No. 2 and No. 3, he does not know anything. It follows from this that the chance of success with the last door will be 66%, i.e. The probability of victory increases twice.

But what will be the situation, if the lead will behave differently?

Paradox Paradox Paradox with a different behavior of the lead

In the classic version of the Monty Hall paradox, it is said that the leading show must necessarily provide the player choosing the door, regardless of whether the player guessed or not. But the lead can and complicate its behavior. For example:

• The host offers a player to change his choice if he initially faithful - the player will always lose if it agrees to change the choice;
• The presenter offers a player to change his choice if he initially did not believe - the player will always win if he agrees;
• The presenter opens the door at random, not knowing what it costs - the chances of the player for winning when changing the door will always be ½;
• The host opens the door with the goat, if the player, really, chose the door with the goat - the chances of the player for winning when the door change will always be ½;
• The presenter always opens the door with goat. If the player chose the door with the machine, the left door with the goat will open with the probability (q) equal to P, and the right - with the probability of q \u003d 1-p. If the presenter opened the door to the left, then the probability of winnings is calculated as 1 / (1 + p). If the presenter opened the door to the right, then: 1 / (1 + q). But the likelihood that the door to the right will be opened, equal to: (1 + q) / 3;
• The conditions from the example above, but p \u003d q \u003d 1/2 - the chances of the player for winning when the door change will always be 2/3;
• The conditions from the example above, but p \u003d 1, and Q \u003d 0. If the presenter opens the door to the right, the change in the choice player will lead to victory, if the door of the left will be opened, the probability of victory will be equal to ½;
• If the lead will always open the door with the goat when the player is chosen the door with a car, and with the probability of ½, if the player is selected the door with the goat, then the chances of the player for winning when changing the door will always be ½;
• If the game is repeated many times, and the car is always on a doorway with the same probability, plus the door opens with the same probability, but the lead knows where the car always puts the player before choosing, opening the door with the goat, the probability of victory will be equal to 1/3;
• The conditions from the example above, but the presenter can not open the door at all - the player's chances of winning will be 1/3.

Such is the paradox of the moon hall. Checking his classic option in practice is quite simple, but it will be much more difficult to carry out experiments with a change in behavior of the master. Although for meticulous practitioners and this is possible. But it does not matter if you will check the Paradox of Monty Hall on personal experience or not, now you know some secrets of games conducted with people on different shows and television shows, as well as interesting mathematical patterns.

By the way, it is interesting: Monti Hall paradox is mentioned in the film Robert Luketich "Twenty-One", the Roman of Sergey Lukyanenko "Nearby", TV series "4), Mark Haddon" Mysterious Night Murder of Dogs ", Kick" XKCD ", and was also a" hero "of one of the TV show series "Destroyers legends."published

Join us in

People are accustomed to consider the right thing that seems obvious. Because they often fall asleep, incorrectly assessing the situation, trusting their intuition and not express time in order to critically comprehend their choice and its consequences.

Monti A visual illustration of a person's inability to weigh his chances of success in choosing a favorable outcome in the presence of more than one unfavorable.

Paradox wording Monti Hall

So, what is this beast? What is actually talking about? The most famous example of the Paradox of Monty Hall is the TV show, popular in America of the middle of the last century called "Let's make a bet!". By the way, it is thanks to the leading of this quiz afterwards and received its name of the Paradox of Monty Hall.

The game was as follows: the participant showed three doors, it seems completely the same. However, at one of them, the player was waiting for an expensive new car, but for two others I was very impatient on the goat. As it usually happens in the case of a televitory, which was behind the chosen by the contestant door, then it became his gain.

What is the trick?

But not everything is so simple. After the choice was made, leading, knowing where the main prize is hidden, opened one of the remaining two doors (of course, the one, the most hidden, and then asked the player, does not want to change his decision.

The Paradox of Monti Hall, formulated by scientists in 1990, is that, contrary to intuition, prompting that there is no difference in making on the basis of a leading decision, you need to agree to change your choice. If you want to get a great car, naturally.

How it works?

The reasons for which people will not want to abandon their choice, several. Intuition and simple (but incorrect) logic say that nothing depends on this solution. Moreover, not everyone wants to go about the other - this is the most real manipulation, isn't it? No not like this. But if everything was immediately intuitive, they would not call. There is nothing strange to doubt. When this puzzle was first published in one of the major magazines, thousands of readers, including recognized mathematicians, sent letters to the editor, in which they argued that the response printed in the room did not correspond to reality. If the existence of probability theory was not news for a person who had fallen on the show, it would be possible to solve this task. And thereby increase the chances of winning. In fact, an explanation of the Paradox of Monty Hall is reduced to a simple mathematics.

The first explanation is more complicated.

The probability that the prize is behind that door, which was originally elected - one of three. The chance to detect it for one of the two remaining is two of the three. Logic, isn't it? Now, after one of these doors turns out to be open, and the goat is found behind it, in the second set (the volume that corresponds to 2/3 of the chance for success) is only one option. The value of this option remains the same, and it is equal to two of three. Thus, it becomes obvious that by changing its decision, the player will increase the probability of winning twice.

Explanation number two, simpler

After such an interpretation of the solution, many still insist that there is no point in this choice, because the option is only two and one of them is exactly winning, and the other definitely leads to defeat.

But the theory of probabilities on this problem is a look. And it becomes even clearer, if you imagine that the doors initially not three, but, say, a hundred. In this case, the opportunity to guess where the prize, the first time is only one to ninety-nine. Now the participant makes his choice, and Monty eliminates ninety-eight doors with goats, leaving only two, one of which chose a player. Thus, the option chosen initially retains the chances of winning equal 1/100, and the second proposed possibility is 99/100. The choice should be obvious.

Are there any refutations?

The answer is simple: no. Not a single reasonable refutation of the Paradox of Monty Hall does not exist. All "exposure", which can be found on the network, are reduced to the lack of understanding of the principles of mathematics and logic.

For everyone who is familiar with the mathematical principles, the probability is absolutely obvious. Do not agree with them can only one who does not understand how logic is arranged. If all of the above still sounds unconvincing - the justification of the paradox has been checked and confirmed on the well-known transmission of "destroyers of the legends", and who else believes, how not to them?

The ability to make sure clearly

Well, let it all sound convincing. But this is only the theory, can I somehow look at the work of this principle in action, and not only in words? First, no one has canceled living people. Find a partner who will take the role of the lead and help play the above-described algorithm in reality. For convenience, you can take boxes, drawers or draw on paper. Repeating the process of several tens of times, compare the number of winnings in case of changing the initial choice with how many victories have brought stubbornness, and everything will become clear. And you can come even easier and use the Internet. There are many simulators of the Paradox Hall paradox simulators, they can be checked all and without too much requisite.

What is the sense of these knowledge?

It may seem that it is just another puzzle, designed to strain brains, and it serves only entertainment purposes. However, its practical application of the Paradox of Monty Hall is primarily in gambling and various tote. Those who have a lot of experience are well known for widespread strategies for increasing the chances of finding a wound bet (from the English word VALUE, which literally means "value" - such a forecast that will come true with a greater probability than it was rated by bookmakers). And one of these strategies directly involves the Monti Hall paradox.

Example in working with a tote

A sports example will differ little from classic. Suppose there are three teams from the first division. In three coming days, each of these teams should play on one decisive match. That of them that, following the results of the match, gain more points than the other two, will remain in the first division, the rest will be forced to leave him. The offer of the bookmaker is simple: you need to put on the preservation of positions of one of these football clubs, while the rates coefficients are equal.

For convenience, such conditions are accepted under which rivals participating in the choice of clubs are approximately equal in force. Thus, it is definitely determined to define the favorite before the start of the games will not work.

Here you need to remember the story about goats and the car. Each of the teams has a chance to stay in its place in one case of three. It is chosen by any of them, a bet is made on it. Let it be "Baltika". According to the results of the first day, one of the clubs loses, and two will only play two. This is the very "Baltika" and, say, "Shinnik".

Most will retain their original bet - "Baltika" will remain in the first division. But it should be remembered that her chances remained the same, but the chances of "shinnik" doubled. Therefore, it is logical to make another bet, larger, on the victory of Shinnik.

The next day comes, and the match with the participation of "Baltika" passes in a draw. The next plays "Shinnik", and his game ends with a victory with a score of 3: 0. It turns out that it is he will remain in the first division. Therefore, at least the first bet on the Baltic and is lost, but this loss overlaps the profit at the new rate on the "Shinnik".

It can be assumed, and the majority will do that the shinnik win is just an accident. In fact, it is likely to take the chance for the accident - the largest mistake for a person participating in sports tote. After all, a professional will always say that any likelihood is expressed primarily in clear mathematical patterns. If you know the foundations of this approach and all the nuances associated with it, then the risks of the loss of money will be minimized.

Benefit in forecasting economic processes

So, in the bets on the sports paradox of Monty Hall to know is just necessary. But the area of \u200b\u200bits use is not limited to one tote. Probability theory is always closely related to statistics, because of politics and economics. Understanding the principles of paradox is equally important.

In the conditions of economic uncertainty, with which analysts often have, it is necessary to remember the following conclusion that follows from solving: it is not necessary to know exactly the only right solution. The chances of a successful forecast are always rising, if you know what exactly will not happen. Actually, this is the most useful conclusion from the Monti Hall paradox.

When the world stands on the threshold of economic shocks, politicians always try to guess the desired action of actions to minimize the consequences of the crisis. Returning to the previous examples, in the field of economy, the task can be described as follows: There are three doors before the heads of countries. One leads to hyperinflation, the second to deflation, and the third to the cherished moderate growth of the economy. But how to find the right answer?

Politicians argue that those or other of their actions will lead to an increase in jobs and an increase in the economy. But leading economists, experienced people, among which even the laureates of the Nobel Prize, clearly demonstrate to them that one of these options will not exactly lead to the desired result. Will there be a choice after this policy? It is extremely unlikely, since in this respect they are not much different from the same TV show participants. Therefore, the probability of error will only increase with increasing the number of advisers.

Is information on the topic exhaust?

In fact, only the "classic" version of the paradox has been considered here, that is, the situation at which the master knows exactly, which of the door is the prize, and only opens the door with the goat. But there are other mechanisms for the behavior of the lead, depending on which the principle of operation of the algorithm and the result of its implementation will differ.

The effect of behavior of the leading paradox

So, what can lead to change the course of events? Let's say different options.

The so-called "Devilish Monty" is a situation in which the host will always offer the player to change his choice, provided that it was originally correct. In this case, the change in the solution will always lead to the defeat.

On the contrary, the "Angelic Monty" is called a similar principle of behavior, but if the player's choice was originally wrong. It is logical that in such a situation a change in the decision will lead to victory.

If the lead opens the doors at random, without having an idea of \u200b\u200bwhat is hidden for each of them, the chances will always be equal to fifty percent. At the same time, a car may be a car behind the opened door.

The lead can 100% open the door with the goat if the player chose a car, and with a 50% probability if the player chose a goat. With this algorithm of actions, if the player changes the choice, it will always be in one case in one case.

When the game is repeated again and again, and the likelihood that the winning will be a certain door will always be arbitrary (as well as what door will open lead, while he knows where the car is hidden, and he always opens the door with the goat and offers to change the choice) - the chance to win will always be equal to one of the three. This is called nash equilibrium.

Equally, as in the same case, but provided that the host is not obliged to open one of the doors at all - the probability of victory will be all equal to 1/3.

While the classic scheme is checked quite easily, experiments with other possible algorithms of behavior of the lead to make much more difficult in practice. But with due dotosity of the experimenter, it is possible.

And yet, what is all this?

Understanding the mechanisms of actions of any logical paradoxes is very useful for a person, his brain and awareness of how the world can actually be arranged, as far as his device may differ from the usual representation of an individual about him.

The more the person knows how the fact that surrounds him in everyday life and what he is not accustomed to think about, the better his consciousness works, and the more effective it can be in his actions and aspirations.