Formulation

The most popular is the task with additional condition No. 6 from the table - the participant in the game knows the following rules in advance:

• the car is equally likely placed behind any of the 3 doors;
• In any case, the presenter is obliged to open the door with the goat and invite the player to change the choice, but not the door that the player chose;
• if the leader has a choice of which of 2 doors to open, he chooses either of them with equal probability.

The following text discusses the Monty Hall problem in precisely this formulation.

Analysis

When solving this problem, they usually reason something like this: the leader always ends up removing one losing door, and then the probability of a car appearing behind two open ones becomes equal to 1/2, regardless of the initial choice.

The whole point is that with his initial choice the participant divides the doors: the chosen one A and two others - B And C. The probability that the car is behind the selected door = 1/3, that it is behind the others = 2/3.

For each of the remaining doors, the current situation is described as follows:

P(B) = 2/3*1/2 = 1/3

P(C) = 2/3*1/2 = 1/3

Where 1/2 is the conditional probability of finding a car exactly behind a given door, provided that the car is not behind the door chosen by the player.

The presenter, opening one of the remaining doors, which is always a losing one, thereby informs the player exactly 1 bit of information and changes the conditional probabilities for B and C, respectively, to “1” and “0”.

As a result, the expressions take the form:

P(B) = 2/3*1 = 2/3

Thus, the participant should change his original choice - in this case, the probability of winning will be equal to 2/3.

One of the simplest explanations is the following: if you change the door after the host's actions, then you win if you initially chose the losing door (then the host will open the second losing one and you will have to change your choice to win). And initially you can choose a losing door in 2 ways (probability 2/3), i.e. if you change the door, you win with a 2/3 probability.

This conclusion contradicts the intuitive perception of the situation by most people, which is why the described task is called Monty Hall paradox, i.e. a paradox in the everyday sense.

And the intuitive perception is this: by opening the door with the goat, the presenter sets a new task for the player, which is in no way connected with the previous choice - after all, the goat will be behind the open door regardless of whether the player previously chose a goat or a car. After the third door is opened, the player will have to make a choice again - and choose either the same door that he chose before, or another. That is, he does not change his previous choice, but makes a new one. The mathematical solution considers two consecutive tasks of the leader as related to each other.

However, one should take into account the factor from the condition that the presenter will open the door with the goat from the remaining two, and not the door chosen by the player. Therefore, the remaining door has a better chance of being the car since it was not selected by the leader. If we consider the case when the presenter, knowing that there is a goat behind the door chosen by the player, nevertheless opens this door, by doing so he will deliberately reduce the player’s chances of choosing the correct door, because probability the right choice it will be already 1/2. But this kind of game will have different rules.

Let's give one more explanation. Let's assume that you play according to the system described above, i.e. of the two remaining doors, you always choose a door different from your original choice. In which case will you lose? A loss will occur if and only if from the very beginning you chose the door behind which the car is located, because subsequently you will inevitably change your decision in favor of the door with a goat, in all other cases you will win, that is, if from the very beginning We made a mistake with the choice of door. But the probability of choosing the door with the goat from the very beginning is 2/3, so it turns out that to win you need an error, the probability of which is twice as high as the correct choice.

Mentions

• In the film Twenty-One, the teacher, Miki Rosa, offers the main character, Ben, to solve a problem: behind three doors there are two scooters and one car, you need to guess the door with the car. After the first choice, Miki suggests changing the choice. Ben agrees and argues mathematically for his decision. So he involuntarily passes the test for Mika’s team.
• In Sergei Lukyanenko’s novel “Klutz”, the main characters, using this technique, win a carriage and the opportunity to continue their journey.
• In the television series “4isla” (episode 13 of season 1 “Man Hunt”), one of the main characters, Charlie Epps, explains the Monty Hall paradox at a popular lecture on mathematics, visually illustrating it using marker boards with goats and a car drawn on the reverse sides. Charlie actually finds the car after changing his choice. However, it should be noted that he is conducting only one experiment, while the advantage of the choice switching strategy is statistical, and a series of experiments should be conducted to properly illustrate it.
• The Monty Hall Paradox is discussed in the diary of the hero of Mark Haddon's story "The Curious Murder of the Dog in the Night-Time."
• The Monty Hall Paradox was tested by MythBusters

See also

• Bertrand's Paradox

Links

	Monty Hall Paradox in Wikibooks
	Monty Hall Paradox on Wikimedia Commons

• Interactive prototype: for those who want to fool around (generation occurs after the first choice)
• Interactive prototype: a real prototype of the game (cards are generated before selection, the work of the prototype is transparent)
• Explanatory video on the website Smart Videos .ru

• Weisstein, Eric W. Monty Hall's Paradox (English) on the Wolfram MathWorld website.
• The Monty Hall Paradox on the website of the TV show Let’s Make a deal
• An excerpt from the book by S. Lukyanenko, which uses the Monty Hall paradox
• Another Bayes solution Another Bayes solution at the Novosibirsk State University forum

Literature

• Gmurman V.E. Probability theory and mathematical statistics, - M.: Higher education. 2005
• Gnedin, Sasha "The Mondee Gills Game." magazine The Mathematical Intelligencer, 2011 http://www.springerlink.com/content/8402812734520774/fulltext.pdf
• Parade Magazine from February 17.
• vos Savant, Marilyn. "Ask Marilyn" column, magazine Parade Magazine from February 26.
• Bapeswara Rao, V. V. and Rao, M. Bhaskara. "A three-door game show and some of its variants." Magazine The Mathematical Scientist, 1992, № 2.
• Tijms, Henk. Understanding Probability, Chance Rules in Everyday Life. Cambridge University Press, New York, 2004. (ISBN 0-521-54036-4)

Notes

Wikimedia Foundation. 2010.

See what the "Monty Hall Paradox" is in other dictionaries:

In search of a car, the player chooses door 1. Then the presenter opens the 3rd door, behind which there is a goat, and invites the player to change his choice to door 2. Should he do this? The Monty Hall paradox is one of the well-known problems of the theory... ... Wikipedia

- (The Tie Paradox) is a well-known paradox, similar to the problem of two envelopes, which also demonstrates the peculiarities of the subjective perception of probability theory. The essence of the paradox: two men give each other ties for Christmas, bought by them... ... Wikipedia

Ecology of knowledge. One of the problems of probability theory is the most interesting and seemingly counterintuitive Monty Hall paradox, named after the host of the American TV show “Let’s Make A Deal.”

Many of us have probably heard about probability theory - a special branch of mathematics that studies patterns in random phenomena, random events, as well as their properties. And just one of the problems of probability theory is the most interesting and seemingly counterintuitive Monty Hall paradox, named after the host of the American TV show “Let’s Make A Deal.” We want to introduce you to this paradox today.

Definition of Monty Hall Paradox

As a problem, the Monty Hall paradox is defined in the form of descriptions of the above-mentioned game, the most common among which is the formulation that was published by Parade Magazine in 1990.

According to it, a person must imagine himself as a participant in a game where he needs to choose one door out of three.

Behind one door is a car, and behind the others are goats. The player must choose one door, for example, door No. 1.

And the leader, who knows what is behind each door, opens one of the two doors that remain, for example, door No. 3, behind which there is a goat.

After this, the host asks the player if he would like to change his original choice and choose door No. 2?

Question: Will a player's chances of winning increase if he changes his choice?

But after the publication of this definition, it turned out that the player’s task was formulated somewhat incorrectly, because All conditions have not been discussed.

For example, the game host may choose a “Monty from Hell” strategy, offering to change the choice only if the player initially guessed the door behind which the car is located.

And it becomes clear that changing the choice will lead to a 100% loss.

Therefore, the most popular formulation of the problem was with special condition No. 6 from a special table:

• A car is equally likely to be behind each door
• The host is always obliged to open the door with a goat other than the one the player has chosen, and offer the player the opportunity to change the choice
• The presenter, having the opportunity to open one of two doors, chooses either one with equal probability

The analysis of the Monty Hall paradox presented below is considered precisely with this condition in mind. So, analysis of the paradox.

Analysis of the Monty Hall Paradox

There are three options for the development of events:

Door 1	Door 2	Door 3	Result if you change the selection	Result if you do not change the choice
Auto	Goat	Goat	Goat	Auto
Goat	Auto	Goat	Auto	Goat
Goat	Goat	Auto	Auto	Goat

When solving the presented problem, the following reasoning is usually given: in each case, the leader removes one door with a goat, therefore, the probability of finding a car behind one of the two closed doors is equal to ½, regardless of what choice was made initially. However, this is not true.

The idea is that by making the first choice, the participant divides the doors into A (selected), B and C (remaining). The chances (P) that the car is behind door A are 1/3, and the chances (P) that it is behind doors B and C are 2/3. And the chances of success when choosing doors B and C are calculated as follows:

P(B) = 2/3 * ½ = 1/3

P(C) = 2/3 * ½ = 1/3

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

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

P(B) = 2/3*1 = 2/3

P(C) = 2/3*0 = 0

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

This is explained as follows: By changing his choice after the leader’s manipulations, the player will win if he initially chose the door with a goat, because the presenter opens the second door with the goat, and the player can only change the doors. There are two ways to initially choose a door with a goat (2/3), respectively, if the player replaces the doors, he will win with a probability of 2/3. It is precisely because this conclusion contradicts intuitive perception that the problem received the status of a paradox.

Intuitive perception suggests the following: when the leader opens the losing door, the player faces new task, at first glance not related to the initial choice, because the goat behind the door opened by the leader will be there in any case, regardless of whether the player initially chose the losing or winning door.

After the leader opens the door, the player must make a choice again - either stay on the previous door or choose a new one. This means that the player makes a new choice, and does not change the original one. AND mathematical solution Two sequential and interconnected tasks of the presenter are considered.

But you need to keep in mind that the presenter opens the door of exactly the two that remain, but not the one the player chose. This means that the chance that the car is behind the remaining door increases, because the presenter did not choose her. If the presenter knows that there is a goat behind the door chosen by the player, he still opens it, thereby obviously reducing the likelihood that the player will choose the right door, because the probability of success will be equal to ½. But this is a game by different rules.

Here's another explanation: Let's say the player plays according to the system presented above, i.e. from doors B or C, he always chooses the one that differs from the original choice. He will lose if he initially chose the door with the car, because subsequently he will choose the door with the goat. In any other case, the player will win if he initially chose the losing option. However, the probability that he will initially choose it is 2/3, which means that to succeed in the game he must first make a mistake, which is twice as likely as the probability of choosing correctly.

Third explanation: Let's imagine that there are not 3 doors, but 1000. After the player has made a choice, the presenter removes 998 unnecessary doors - only two doors remain: the one chosen by the player and one more. But the chance that there is a car behind each door is not at all ½. Most likely (0.999%) the car will be behind the door that the player did not initially choose, i.e. behind the door selected from the 999 others remaining after the first choice. You need to think about the same when choosing from three doors, even if the chances of success decrease and become 2/3.

AND last explanation– replacement of conditions. Let's say that instead of making an initial choice of, say, door #1, and instead of having the host open door #2 or #3, the player must make the right choice the first time if he knows that the probability of success with door #1 is 33 %, but he knows nothing about the absence of a car behind doors No. 2 and No. 3. It follows from this that the chance of success with the last door will be 66%, i.e. the probability of winning doubles.

But what will the state of affairs be if the presenter behaves differently?

Analysis of the Monty Hall paradox with a different behavior of the presenter

The classic version of the Monty Hall Paradox states that the show's host must always give the player a choice of door, regardless of whether the player guessed right or not. But the leader can also complicate his behavior. For example:

• The presenter invites the player to change his choice if it is initially correct - the player will always lose if he agrees to change the choice;
• The presenter invites the player to change his choice if it is initially incorrect - the player will always win if he agrees;
• The presenter opens the door at random, not knowing what is where - the player’s chances of winning when changing the door will always be ½;
• The presenter opens the door with a goat, if the player actually chose the door with a goat, the player’s chances of winning when changing the door will always be ½;
• The host always opens the door with a goat. If the player chose the door with the car, the left door with the goat will open with probability (q) equal to p, and the right door with probability q = 1-p. If the leader opened the door on the left, then the probability of winning is calculated as 1/(1+p). If the leader opened the door on the right, then: 1/(1+q).But the probability that the door on the right will be opened is: (1+q)/3;
• Conditions from the example above, but p=q=1/2 - the player’s chances of winning when changing the door will always be 2/3;
• Conditions from the example above, but p=1 and q=0. If the leader opens the door on the right, then the player’s change of choice will lead to victory, if the door on the left is opened, then the probability of victory will be equal to ½;
• If the host always opens the door with a goat when the player chooses the door with the car, and with probability ½ if the player chooses the door with the goat, then the player's chances of winning when changing the door will always be ½;
• If the game is repeated many times, and the car is always behind one or another door with the same probability, plus the leader opens the door with the same probability, but the leader knows where the car is and always puts the player before a choice, opening the door with a goat, then the probability of winning will be equal 1/3;
• The conditions are from the example above, but the presenter may not open the door at all - the player’s chances of winning will be 1/3.

This is the Motney Hall paradox. Check it out classic version in practice it is quite simple, but it will be much more difficult to conduct experiments with changing the behavior of the presenter. Although for meticulous practitioners this is possible. But it doesn't matter whether you test the Monty Hall Paradox for personal experience or not, now you know some of the secrets of games played with people on different shows and TV shows, as well as interesting mathematical patterns.

By the way, this is interesting: The Monty Hall paradox is mentioned in Robert Luketic's film "Twenty-One", Sergei Lukyanenko's novel "The Klutz", the television series "4isla", the story by Mark Haddon "The Mysterious Murder of a Dog in the Night-Time", the comic book "XKCD", and was also the "hero" of one of the episodes of the television show "MythBusters" published

Join us on

People are accustomed to considering what seems obvious to be correct. That’s why they often get into trouble by misjudging the situation, trusting their intuition, and not taking the time to critically think about their choices and their consequences.

Monty is a clear illustration of a person's inability to weigh their chances of success when choosing a favorable outcome in the presence of more than one unfavorable one.

Formulation of the Monty Hall Paradox

So, what kind of animal is this? What exactly are we talking about? The most famous example The Monty Hall paradox is represented by a television show popular in America in the middle of the last century called “Let's Make a Bet!” By the way, it was thanks to the host of this quiz that the Monty Hall Paradox subsequently received its name.

The game consisted of the following: the participant was shown three doors that looked exactly the same. However, behind one of them there was a road waiting for the player new car, but the other two were impatiently pining for the goat. As is usually the case with game shows, whatever was behind the contestant's chosen door became his winnings.

What is the trick?

But it's not that simple. After the choice was made, the presenter, knowing where the main prize was hidden, opened one of the remaining two doors (of course, the one behind which the artiodactyl was hiding), and then asked the player if he would like to change his decision.

The Monty Hall paradox, formulated by scientists in 1990, is that, contrary to intuition that there is no difference in making a leading decision based on an issue, one must agree to change one's choice. If you want to get a great car, of course.

How does this work?

There are several reasons why people will not want to give up their choice. Intuition and simple (but incorrect) logic say that nothing depends on this decision. Moreover, not everyone wants to follow the lead of another - this is real manipulation, isn’t it? No, not like that. But if everything was immediately intuitive, they wouldn’t even name it. It's not strange to have doubts. When this puzzle was first published in one of the major magazines, thousands of readers, including recognized mathematicians, sent letters to the editor claiming that the answer printed in the issue was not true. If the existence of probability theory was not news to the person who got on the show, then perhaps he would be able to solve this problem. And thereby increase the chances of winning. In fact, the explanation of the Monty Hall paradox comes down to simple mathematics.

Explanation one, more complicated

The probability that the prize is behind the door that was originally chosen is one in three. The chance of finding it behind one of the remaining two is equal to two out of three. Logical, right? Now, after one of these doors is opened and a goat is found behind it, there is only one option left in the second set (the one that corresponds to the 2/3 chance of success). The value of this option remains the same, which is two out of three. Thus, it becomes obvious that by changing his decision, the player will double the probability of winning.

Explanation number two, simpler

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

But probability theory has this problem your opinion. And this becomes even clearer if we imagine that there were initially not three doors, but, say, a hundred. In this case, it is possible to guess where the prize, the first time, is only one in ninety-nine. Now the participant makes his choice, and Monty eliminates ninety-eight doors with goats, leaving only two, one of which the player chose. Thus, the option chosen initially retains the odds of winning equal to 1/100, and the second option offered remains 99/100. The choice should be obvious.

Are there any refutations?

The answer is simple: no. There is not a single sufficiently substantiated refutation of the Monty Hall paradox. All the “revelations” that can be found on the Internet boil down to a misunderstanding of the principles of mathematics and logic.

For anyone who is well acquainted with mathematical principles, the non-randomness of probabilities is absolutely obvious. Only those who do not understand how logic works can disagree with them. If all of the above still sounds unconvincing, the rationale for the paradox was tested and confirmed on the famous program “MythBusters,” and who else to believe if not them?

Possibility to see clearly

Okay, let all this sound convincing. But this is only a theory, is it possible to somehow look at the work of this principle in action, and not just in words? Firstly, no one canceled living people. Find a partner who will take on the role of facilitator and help you play out the algorithm described above in reality. For convenience, you can take boxes, crates, or even draw on paper. After repeating the process several dozen times, compare the number of wins in case of changing the initial choice with how many wins brought by stubbornness, and everything will become clear. Or you can do it even simpler and use the Internet. There are many simulators of the Monty Hall paradox on the Internet, in which you can test everything yourself and without unnecessary props.

What is the use of this knowledge?

It may seem that this is just another puzzle designed to strain your brain, and it serves only entertainment purposes. However, its practical application The Monty Hall paradox is found primarily in gambling and various sweepstakes. Those with extensive experience are well aware of common strategies for increasing the chances of finding a value bet (from English word value, which literally means “value” - a forecast that is more likely to come true than it was estimated by bookmakers). And one of these strategies directly involves the Monty Hall Paradox.

Example in working with betting

A sports example will differ little from a classic one. Let's say there are three teams from the first division. In the next three days, each of these teams must play one decisive match. The one that scores more points than the other two at the end of the match will remain in the first division, while the rest will be forced to leave it. The bookmaker's offer is simple: you need to bet on maintaining the position of one of these football clubs, while the betting odds are equal.

For convenience, conditions are accepted under which the rivals of the clubs participating in the selection are approximately equal in strength. Thus, it will not be possible to clearly determine the favorite before the start of the games.

Here you need to remember the story about goats and a car. Each team has a one in three chance of staying in its place. Any of them is selected and a bet is placed on it. Let it be Baltika. According to the results of the first day, one of the clubs loses, and two have yet to play. This is the same “Baltika” and, say, “Shinnik”.

The majority will retain their original bid - Baltika will remain in the first division. But it should be remembered that its chances remained the same, but Shinnik’s chances doubled. Therefore, it is logical to make another bet, a larger one, on Shinnik’s victory.

The next day comes, and the match involving Baltika ends in a draw. Shinnik plays next, and their game ends with a victory with a score of 3:0. It turns out that he will remain in the first division. Therefore, although the first bet on Baltika is lost, this loss is covered by the profit on the new bet on Shinnik.

One can assume, and most will do so, that Shinnik's win was just an accident. In fact, mistaking probability for chance is the biggest mistake for a person participating in sports betting. After all, a professional will always say that any probability is expressed primarily in clear mathematical patterns. If you know the basics of this approach and all the nuances associated with it, then the risks of losing money will be minimized.

Useful in forecasting economic processes

So, in sports betting, the Monty Hall paradox is simply necessary to know. But its scope of application is not limited to betting. Probability theory is always closely related to statistics, which is why understanding the principles of paradox is no less important in politics and economics.

In conditions of economic uncertainty, which analysts often deal with, one must remember the following conclusion arising from solving the problem: it is not necessary to know exactly the only correct solution. The chances of a successful forecast always increase if you know what will definitely not happen. Actually, this is the most useful conclusion from the Monty Hall paradox.

When the world is on the verge of economic turmoil, politicians always try to guess the right course of action in order to minimize the consequences of the crisis. Returning to previous examples, in the economic sphere the task can be described as follows: there are three doors before the leaders of countries. One leads to hyperinflation, the second to deflation, and the third to the cherished moderate economic growth. But how to find the right answer?

Politicians claim that their actions will lead to more jobs and economic growth. But leading economists, experienced people, including even laureates Nobel Prize, clearly demonstrate to them that one of these options will definitely not lead to desired result. Will politicians change their choices after this? It is extremely unlikely, since in this respect they are not much different from the same participants in the television show. Therefore, the likelihood of error will only increase as the number of advisors increases.

Does this exhaust the information on the topic?

In fact, so far only the “classic” version of the paradox has been considered here, that is, the situation in which the presenter knows exactly which door is behind the prize, and only opens the door with the goat. But there are other mechanisms of the leader’s behavior, depending on which the principle of operation of the algorithm and the result of its execution will differ.

The influence of the leader's behavior on the paradox

So what can the presenter do to change the course of events? Let's allow different options.

The so-called “Devil Monty” is a situation in which the host will always offer the player to change his choice, provided that it was initially correct. In this case, changing the decision will always lead to defeat.

On the contrary, “Angel Monty” is a similar principle of behavior, but in the event that the player’s choice was initially incorrect. It is logical that in such a situation changing the decision will lead to victory.

If the presenter opens the doors at random, having no idea what is hidden behind each of them, then the chances of winning will always be fifty percent. In this case, there may be a car behind the open leading door.

The GM has a 100% chance of opening the door with a goat if the player chose a car, and with a 50% chance if the player chose a goat. With this algorithm of actions, if the player changes his choice, he will always win in one case out of two.

When the game is repeated over and over again, and the probability of a particular door winning is always arbitrary (as well as which door the presenter will open, while he knows where the car is hiding, and he always opens the door with a goat and offers to change the choice) - the chance of winning will always be equal to one in three. This is called a Nash equilibrium.

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

While the classic scheme is quite easy to test, experiments with other possible behavior algorithms for the presenter are much more difficult to carry out in practice. But with due meticulousness of the experimenter, this is also possible.

And yet, what is all this for?

Understanding the mechanisms of action of any logical paradoxes very useful for a person, his brain and awareness of how the world can actually be structured, how much its structure can differ from the individual’s usual idea of ​​it.

How more people knows how things around him work everyday life and what he is not at all used to thinking about, the better his consciousness works, and the more effective he can be in his actions and aspirations.

About lotteries

This game has long become widespread and has become an integral part of modern life. And although the lottery is increasingly expanding its capabilities, many people still see it only as a way to get rich. It may not be free or reliable. On the other hand, as one of Jack London's heroes noted, in gambling You can’t ignore the facts - sometimes people get lucky.

Mathematics of chance. History of probability theory

Alexander Bufetov

Transcript and video recording of the lecture by Doctor of Physical and Mathematical Sciences, presenter research fellow Steklov Mathematical Institute, leading researcher at the Institute of Industrial Problems of the Russian Academy of Sciences, professor at the Faculty of Mathematics High school economics, research director National Center scientific research in France (CNRS) by Alexander Bufetov, given as part of the series “Public lectures “Polit.ru”” on February 6, 2014.

The illusion of regularity: why randomness seems unnatural

Our ideas about the random, the natural and the impossible often diverge from the data of statistics and probability theory. In the book “Imperfect Chance. How chance rules our lives,” American physicist and science popularizer Leonard Mlodinow talks about why random algorithms look so strange, what the catch is in “randomly” shuffling songs on an iPod, and what a stock analyst’s luck depends on. “Theories and Practices” publishes an excerpt from the book.

Determinism

Determinism is a general scientific concept and philosophical doctrine about causality, patterns, genetic connections, interaction and conditionality of all phenomena and processes occurring in the world.

God is a statistic

Deborah Nolan, professor of statistics at the University of California at Berkeley, asks her students to complete a very strange task at first glance. The first group must toss a coin a hundred times and write down the result: heads or tails. The second must imagine that she is tossing a coin - and also make a list of hundreds of “imaginary” results.

What is determinism

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

The Picky Bride Problem

Huseyn-Zade S. M.

Zeno's paradox

Is it possible to get from one point in space to another? The ancient Greek philosopher Zeno of Elea believed that movement could not be accomplished at all, but how did he argue for this? Colm Keller will talk about how to resolve the famous Zeno's paradox.

Paradoxes of infinite sets

Imagine a hotel with an infinite number of rooms. A bus arrives with an endless number of future guests. But placing them all is not so easy. This is an endless hassle, and the guests are endlessly tired. And if you fail to cope with the task, then you can lose an infinite amount of money! What to do?

Dependence of child's height on parents' height

Young parents, of course, want to know how tall their child will be as an adult. Mathematical statistics can offer a simple linear relationship to approximate the height of children based only on the height of the father and mother, and also indicate the accuracy of such an estimate.

Monty Hall's paradox is probably the most famous paradox in probability theory. There are many variations of it, for example, the paradox of three prisoners. And there are many interpretations and explanations of this paradox. But here, I would like to give not only a formal explanation, but show the “physical” basis of what happens in the Monty Hall paradox and others like it.

The classic formulation is:

“You are a participant in the game. There are three doors in front of you. There's a prize for one of them. The host invites you to try to guess where the prize is. You point to one of the doors (at random).

Formulation of the Monty Hall Paradox

The host knows where the prize actually is. He doesn’t yet open the door you pointed to. But it opens one more of the remaining doors for you, behind which there is no prize. The question is, should you change your choice or stay with your previous decision?

It turns out that if you simply change your choices, your chances of winning will increase!

The paradox of the situation is obvious. It seems that everything that happens is random. It makes no difference whether you change your mind or not. But that's not true.

"Physical" explanation of the nature of this paradox

Let's first not go into mathematical subtleties, but simply look at the situation with an open mind.

In this game you only do first random selection. Then the presenter tells you additional information , which allows you to increase your chances of winning.

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

Let's, for the sake of simplicity (although there is an element of deceit in this), consider a more likely situation: you pointed to a door behind which there is no prize. Then, behind one of the remaining doors is a prize There is. That is, the presenter has no choice. He opens a very specific door. (You pointed to one, there is a prize behind the other, there is only one door left that the leader can open.)

It is at this moment of meaningful choice that he gives you information that you can use.

IN in this case, the use of information is that you change your decision.

By the way, your second choice is already too not accidental(or rather, not as random as the first choice). After all, you choose from closed doors, but one is already open and it not arbitrary.

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

A more formal explanation of the Monty Hall paradox

In fact, your first, random choice splits all the doors into two groups. Behind the door you chose there is a prize with a probability of 1/3, behind the other two - with a probability of 2/3. Now the leader makes a change: he opens one door in the second group. And now the entire 2/3 probability only applies to closed door from a group of two doors.

It is clear that now it is more profitable for you to change your decision.

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

However, changing your selection increases your chances of winning.

Monty Hall Paradox

The Monty Hall paradox is a probabilistic problem, the solution of which (according to some) is contrary to common sense. Problem formulation:

Imagine that you are a participant in a game in which you need to choose one of three doors. Behind one of the doors is a car, behind the other two doors are goats.
You choose one of the doors, for example, number 1, after which the leader, who knows where the car is and where the goats are, opens one of the remaining doors, for example, number 3, behind which there is a goat.

Monty Hall paradox. The most inaccurate mathematics

He then asks you if you would like to change your choice and choose door number 2.
Will your chances of winning a car increase if you accept the presenter's offer and change your choice?

When solving a problem, it is often mistakenly assumed that the two choices are independent and, therefore, the probability will not change if the choice is changed. In fact, this is not the case, as you can see by remembering Bayes' formula or looking at the simulation results below:

Here: “strategy 1” - do not change the choice, “strategy 2” - change the choice. Theoretically, for the case with 3 doors, the probability distribution is 33.(3)% and 66.(6)%. Numerical simulations should yield similar results.

Links

Monty Hall Paradox– a problem from the section of probability theory, the solution of which contradicts common sense.

History[edit | edit wiki text]

At the end of 1963, a new talk show called “Let’s Make a Deal” aired. According to the quiz script, viewers from the audience received prizes for correct answers, having a chance to increase them by making new bets, but risking their existing winnings. The show's founders were Stefan Hatosu and Monty Hall, the latter of whom became its constant host for many years.

One of the tasks for the participants was the drawing of the Main Prize, which was located behind one of three doors. Behind the remaining two were incentive prizes, and the presenter, in turn, knew the order of their arrangement. The contestant had to determine the winning door by betting their entire winnings for the show.

When the guesser decided on the number, the presenter opened one of the remaining doors, behind which there was an incentive prize, and invited the player to change the initially chosen door.

Wording[edit | edit wiki text]

As a specific problem, the paradox was first formulated by Steve Selvin in 1975, when he sent The American Statistician magazine and host Monty Hall the question: would a contestant's chances of winning the Grand Prize change if, after opening the door with incentive will he change his choice? After this incident, the concept of the “Monty Hall Paradox” appeared.

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

“Imagine yourself on a game show where you have to choose one of three doors: two of them are goats, and the third is a car. When you make a choice, assuming, for example, that the winning door is number one, the leader opens one of the remaining two doors, for example, number three, behind which is a goat. Then you are given a chance to change the selection to another door? Can you increase your chances of winning a car if you change your choice from door number one to door number two?

This formulation is a simplified version, because There remains the factor of influence of the presenter, who knows exactly where the car is and is interested in the participant’s loss.

For the task to become purely mathematical, it is necessary to eliminate the human factor by introducing the opening of a door with an incentive prize and the ability to change the initial choice as integral conditions.

Solution[edit | edit wiki text]

When comparing chances, at first glance, changing the door number will not give any advantages, because all three options have a 1/3 chance of winning (approx. 33.33% for each of the three doors). In this case, opening one of the doors will not in any way affect the chances of the remaining two, whose chances will become 1/2 to 1/2 (50% for each of the two remaining doors). This judgment is based on the assumption that the player’s choice of a door and the leader’s choice of a door are two independent events that do not affect one another. In reality, it is necessary to consider the entire sequence of events as a whole. In accordance with the theory of probability, the chances of the first selected door from the beginning to the end of the game are invariably 1/3 (approx. 33.33%), and the two remaining ones have a total of 1/3+1/3 = 2/3 (approx. 66.66%). When one of the two remaining doors opens, its chances become 0% (there is an incentive prize hidden behind it), and as a result, the chances of closing the unselected door will be 66.66%, i.e. twice as much as the one originally selected.

To make it easier to understand the results of a choice, you can consider an alternative situation in which the number of options will be greater, for example, a thousand. The probability of choosing the winning option is 1/1000 (0.1%). Given that nine hundred and ninety-eight incorrect ones are subsequently opened out of the remaining nine hundred and ninety-nine options, it becomes clear that the probability of the one remaining door out of the nine hundred and ninety-nine not chosen is higher than that of the only one chosen at the beginning.

Mentions[edit | edit wiki text]

You can find references to the Monty Hall Paradox in “Twenty-One” (a film by Robert Luketic), “The Klutz” (a novel by Sergei Lukyanenko), the television series “4isla” (TV series), “The Mysterious Murder of a Dog in the Night-Time” (a story by Mark Haddon), “XKCD” ( comic book), “MythBusters” (TV show).

See also[edit | edit wiki text]

The image shows the process of choosing between two buried doors from the three initially proposed

Examples of solutions to combinatorics problems

Combinatorics is a science that everyone encounters in everyday life: how many ways to choose 3 people on duty to clean the classroom or how many ways to form a word from given letters.

In general, combinatorics allows you to calculate how many different combinations, according to certain conditions, can be made from given objects (same or different).

As a science, combinatorics originated in the 16th century, and now every student (and often even schoolchildren) studies it. They start studying with the concepts of permutations, placements, combinations (with or without repetitions), you will find problems on these topics below. The most well-known rules of combinatorics are the sum and product rules, which are most often used in typical combinatorial problems.

Below you will find several examples of problems with solutions using combinatorial concepts and rules that will help you understand typical tasks. If you have difficulties with the tasks, order a test on combinatorics.

Combinatorics problems with online solutions

Task 1. Mom has 2 apples and 3 pears. Every day for 5 days in a row she gives out one fruit. In how many ways can this be done?

Solution of combinatorics problem 1 (pdf, 35 Kb)

Task 2. An enterprise can provide work for 4 women in one specialty, 6 men for another, and 3 workers for a third, regardless of gender. In how many ways can vacancies be filled if there are 14 applicants: 6 women and 8 men?

Solution of problem in combinatorics 2 (pdf, 39 Kb)

Task 3. There are 9 carriages in a passenger train. In how many ways can 4 people be seated on a train, provided that they all travel in different carriages?

Solution of combinatorics problem 3 (pdf, 33 Kb)

Task 4. There are 9 people in the group. How many different subgroups can you form, provided that the subgroup includes at least 2 people?

Solution to combinatorics problem 4 (pdf, 34 Kb)

Task 5. A group of 20 students needs to be divided into 3 teams, and the first team should include 3 people, the second - 5 and the third - 12. In how many ways can this be done?

Solution of problem in combinatorics 5 (pdf, 37 Kb)

Task 6. The coach selects 5 boys out of 10 to be on the team. In how many ways can he form the team if 2 specific boys are to be on the team?

Combinatorics problem with solution 6 (pdf, 33 Kb)

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

Combinatorics problem with solution 7 (pdf, 37 Kb)

Task 8. How many different fractions can be made from the numbers 3, 5, 7, 11, 13, 17 so that each fraction contains 2 different numbers? How many of them are proper fractions?

Combinatorics problem with solution 8 (pdf, 32 Kb)

Task 9. How many words can you get by rearranging the letters in the word Mountain and Institute?

Combinatorics problem with solution 9 (pdf, 32 Kb)

Problem 10. Which numbers from 1 to 1,000,000 are greater: those in which the unit occurs, or those in which it does not occur?

Combinatorics problem with solution 10 (pdf, 39 Kb)

Ready-made examples

Need solved combinatorics problems? Find in the workbook:

Other solutions to problems in probability theory

“There are three kinds of lies: lies, damned lies and statistics.” This phrase, attributed by Mark Twain to British Prime Minister Benjamin Disraeli, fairly reflects the attitude of the majority towards mathematical laws. Indeed, probability theory sometimes throws up amazing facts, which are difficult to believe at first glance - and which, nevertheless, are confirmed by science. “Theories and Practices” recalled the most famous paradoxes.

Monty Hall Problem

This is exactly the problem that a cunning MIT professor presented to students in the movie Twenty-One. Having given the correct answer, main character ends up on a team of brilliant young mathematicians beating casinos in Las Vegas.

The classic formulation goes like this: “Let’s say a certain player is offered to take part in the famous American TV show Let’s Make a Deal, hosted by Monty Hall, and he needs to choose one of three doors. Behind two doors are goats, behind one is the main prize, a car, the presenter knows the location of the prizes. After the player makes his choice, the host opens one of the remaining doors, behind which there is a goat, and invites the player to change his decision. Should the player agree or is it better to keep his original choice?”

Here's a typical line of reasoning: after the host has opened one of the doors and shown the goat, the player has to choose between two doors. The car is located behind one of them, which means that the probability of guessing it is ½. So it makes no difference whether to change your choice or not. And yet, probability theory says that you can increase your chances of winning by changing your decision. Let's figure out why this is so.

To do this, let's take a step back. The moment we made our initial choice, we divided the doors into two parts: the one we chose and the other two. Obviously, the probability that the car is hiding behind “our” door is ⅓ - accordingly, the car is behind one of the two remaining doors with a probability of ⅔. When the presenter shows that there is a goat behind one of these doors, it turns out that this ⅔ chance falls on the second door. And this reduces the player’s choice to two doors, behind one of which (initially selected) the car is located with a probability of ⅓, and behind the other - with a probability of ⅔. The choice becomes obvious. Which, of course, does not change the fact that from the very beginning the player could choose the door with the car.

Three Prisoners Problem

The Three Prisoners Paradox is similar to the Monty Hall problem, although it takes place in a more dramatic setting. Three prisoners (A, B and C) are sentenced to death penalty and placed in solitary confinement. The governor randomly selects one of them and gives him a pardon. The warden knows which of the three has been pardoned, but he is ordered to keep it a secret. Prisoner A asks the guard to tell him the name of the second prisoner (besides himself) who will definitely be executed: “if B is pardoned, tell me that C will be executed. If B is pardoned, tell me that B will be executed. If they are both executed , and I have been pardoned, toss a coin and say any of these two names.” The warden says that prisoner B will be executed. Should prisoner A be happy?

It would seem so. After all, before receiving this information, the probability of prisoner A’s death was ⅔, and now he knows that one of the other two prisoners will be executed - which means that the probability of his execution has decreased to ½. But in fact, prisoner A did not learn anything new: if he was not pardoned, he would be told the name of another prisoner, and he already knew that one of the two remaining would be executed. If he is lucky and the execution is canceled, he will hear a random name B or C. Therefore, his chances of salvation have not changed in any way.

Now let’s imagine that one of the remaining prisoners finds out about prisoner A’s question and the answer he received. This will change his views on the likelihood of a pardon.

If prisoner B overheard the conversation, he will know that he will definitely be executed. And if prisoner B, then the probability of his pardon will be ⅔. Why did this happen? Prisoner A has not received any information and still has a ⅓ chance of being pardoned. Prisoner B will definitely not be pardoned, and his chances are zero. This means that the probability that the third prisoner will be released is ⅔.

Paradox of two envelopes

This paradox became known thanks to the mathematician Martin Gardner, and is formulated as follows: “Suppose you and a friend were offered two envelopes, one of which contains a certain amount of money X, and the other contains an amount twice as much. You independently open the envelopes, count the money, and then you can exchange them. The envelopes are the same, so the probability that you will receive an envelope with a lower amount is ½. Let's say you open an envelope and find $10 in it. Therefore, it is equally likely that your friend's envelope contains $5 or $20. If you decide to exchange, then you can calculate the mathematical expectation of the final amount - that is, its average value. It is 1/2x$5+1/2x20=$12.5. Thus, the exchange is beneficial to you. And, most likely, your friend will think the same way. But it is obvious that the exchange cannot be beneficial for both of you. What is the mistake?

The paradox is that until you open your envelope, the probabilities behave well: you actually have a 50% chance of finding the amount X in your envelope and a 50% chance of finding the amount 2X. And common sense dictates that information about the amount you have cannot affect the contents of the second envelope.

However, as soon as you open the envelope, the situation changes dramatically (this paradox is somewhat similar to the story of Schrödinger's cat, where the very presence of an observer affects the state of affairs). The fact is that in order to comply with the conditions of the paradox, the probability of finding in the second envelope a larger or smaller amount than yours must be the same. But then any value of this sum from zero to infinity is equally probable. And if there is an equally probable infinite number of possibilities, they add up to infinity. And this is impossible.

For clarity, you can imagine that you find one cent in your envelope. Obviously, the second envelope cannot contain half the amount.

It is curious that discussions regarding the resolution of the paradox continue to this day. At the same time, attempts are being made both to explain the paradox from the inside and to develop the best strategy behavior in such a situation. In particular, Professor Thomas Cover proposed an original approach to strategy formation - to change or not to change the envelope, guided by some intuitive expectation. Let's say, if you open an envelope and find $10 in it - a small amount in your estimation - it's worth exchanging it. And if there is, say, $1,000 in the envelope, which exceeds your wildest expectations, then there is no need to change. This intuitive strategy, if you are regularly asked to choose two envelopes, allows you to increase your total winnings more than the strategy of constantly changing envelopes.

Boy and girl paradox

This paradox was also proposed by Martin Gardner and is formulated as follows: “Mr. Smith has two children. At least one child is a boy. What is the probability that the second one is also a boy?

It would seem that the task is simple. However, if you start to look into it, a curious circumstance emerges: the correct answer will differ depending on how we calculate the probability of the gender of the other child.

Option 1

Let's consider all possible combinations in families with two children:

Girl/Girl

Girl/Boy

Boy/Girl

Boy/Boy

The girl/girl option does not suit us according to the conditions of the task. Therefore, for Mr. Smith's family, there are three equally probable options - which means that the probability that the other child will also be a boy is ⅓. This is exactly the answer that Gardner himself initially gave.

Option 2

Let's imagine that we meet Mr. Smith on the street when he is walking with his son. What is the probability that the second child is also a boy? Since the gender of the second child has nothing to do with the gender of the first, the obvious (and correct) answer is ½.

Why is this happening, since it would seem that nothing has changed?

It all depends on how we approach the issue of calculating probability. In the first case we considered everything possible options Smith family. In the second, we considered all families falling under prerequisite"there must be one boy." The calculation of the probability of the sex of the second child was carried out with this condition (in probability theory this is called “conditional probability”), which led to a result different from the first.