Monty Hall paradox when 2 is greater than 3. Monty Hall paradox: formulation and explanation

The solution of which, at first glance, contradicts common sense.

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  The problem is formulated as a description of a game based on the American game show Let's Make a Deal, and is named after the host of that show. The most common formulation of this problem, published in 1990 in the journal Parade Magazine, sounds like this:

  Imagine that you have become a participant in a game in which you need to choose one of three doors. Behind one of the doors there is a car, behind the other two doors there 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. After that, he 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?

  After publication, it immediately became clear that the task was formulated incorrectly: not all conditions were specified. For example, the presenter may follow the “Monty from Hell” strategy: offer a change of choice if and only if the player chose a car as their first move. Obviously, changing the initial choice will lead to a guaranteed loss in such a situation (see below).

  The most popular is a task with an additional condition - the participant in the game knows the following rules in advance:

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

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

  Analysis

  For the winning strategy, the following is important: if you change the choice of door after the actions of the leader, then you win if you initially chose the losing door. This is likely to happen 2 ⁄ 3 , since initially you can choose a losing door in 2 out of 3 ways.

  But often when solving this problem, they reason something like this: the leader always ends up removing one losing door, and then the probability of a car appearing behind two not open ones becomes equal to ½, regardless of the initial choice. But this is not true: although there are indeed two possibilities for choice, these possibilities (taking into account the background) are not equally probable! This is true because all doors initially had an equal chance of winning, but then had different probabilities of being eliminated.

  For most people, this conclusion contradicts the intuitive perception of the situation, and due to the resulting discrepancy between the logical conclusion and the answer to which the intuitive opinion inclines, the problem is called Monty Hall paradox.

  The situation with doors becomes even more clear if you imagine that there are not 3 doors, but, say, 1000, and after the player’s choice, the presenter removes the extra 998, leaving 2 doors: the one the player chose and one more. It seems more obvious that the probabilities of finding a prize behind these doors are different and not equal to ½. A much higher probability of finding it, namely 0.999, will occur when changing the decision and choosing a door selected from 999. In the case of 3 doors, the logic remains the same, but the probability of winning when changing the decision is lower, namely 2 ⁄ 3 .

  Another way of reasoning is to replace the condition with an equivalent one. Let's imagine that instead of the player making the initial choice (let it always be door No. 1) and then the leader opening the door with the goat among the remaining ones (that is, always among No. 2 and No. 3), imagine that the player needs to guess the door on the first try, but he is previously informed that there may be a car behind door No. 1 with an initial probability (33%), and among the remaining doors it is indicated which of the doors there is definitely no car behind (0%). Accordingly, the last door will always account for 67%, and the strategy for choosing it is preferable.

  Other presenter behavior

  The classic version of the Monty Hall paradox states that the host will definitely offer the player to change the door, regardless of whether he chose the car or not. But more complex behavior of the leader is also possible. This table briefly describes several behaviors.

  Possible behavior of the presenter
  Presenter behavior	Result
  "Hell Monty": The host suggests changing if the door is right.	A change will always produce a goat.
  "Angel Monty": the host suggests changing if the door is wrong.	A change will always give you a car.
  “Ignorant Monty” or “Monty Buh”: the presenter accidentally falls, the door opens, and it turns out that there is no car behind it. In other words, the presenter himself does not know what is behind the doors, he opens the door completely at random, and only by chance there was no car behind it.	The change gives a gain in ½ of the cases. This is exactly how the American show “Deal or No Deal” works - however, a random door is opened by the player himself, and if there is no car behind it, the host offers to change it.
  The host chooses one of the goats and opens it if the player chose another door.	The change gives a gain in ½ of the cases.
  The leader always opens the goat. If a car is selected, the left goat opens with the probability p and right with probability q=1−p.	If the leader opened the left door, the shift gives a win with the probability 1 1 + p (\displaystyle (\frac (1)(1+p))). If right - 1 1 + q (\displaystyle (\frac (1)(1+q))). However, the subject cannot in any way influence the probability that the right door will be opened - regardless of his choice, this will happen with probability 1 + q 3 (\displaystyle (\frac (1+q)(3))).
  The same, p=q= ½ (classical case).	The change gives a win with probability 2 ⁄ 3 .
  The same, p=1, q=0 (“powerless Monty” - the tired presenter stands at the left door and opens the goat that is closer).	If the leader opens the right door, the change gives a guaranteed win. If left - probability ½.
  The presenter always opens the goat if a car is chosen, and with a probability of ½ otherwise.	The change gives a win with a probability of ½.
  General case: the game is repeated many times, the probability of hiding a car behind one or another door, as well as opening one or another door is arbitrary, but the leader knows where the car is and always offers a change, opening one of the goats.	Nash equilibrium: the leader benefits most from the Monty Hall paradox in its classical form (probability of winning 2 ⁄ 3 ). The car hides behind any of the doors with probability ⅓; if there is a choice, we open any goat at random.
  The same thing, but the presenter may not open the door at all.	Nash equilibrium: it is profitable for the leader not to open the door, the probability of winning is ⅓.

  see also

  Notes

  1. Tierney, John (July 21, 1991), "Behind Monty's Hall"s Doors: Puzzle, Debate and Answer? ", The New York Times, . Retrieved January 18, 2008.

  The Monty Hall Paradox began to appear more and more often on bookmaker websites. What is it and can the player use it to his advantage?

  What is Monty Hall's Paradox?

  

  Monty Hall's paradox is a problem from probability theory. It gained its popularity thanks to an American television show where the player has to open one of three doors. Naturally, the prize is behind only one door (a car), and behind the other two is a goat (the show, after all). First the player selects a door. It doesn't open yet. There are two doors left. Of these two doors, the leader must open the one behind which there is a goat. As a result, there are two unopened doors left, one of which is the one the player chose. Behind one is a goat, behind the other is a car. The host offers the player to change his initial choice and open another door. What happens to a player's chances of winning a prize if he changes his mind, and is there any point in doing so?

  If the player changes his choice, he wins with a probability of 66.6%. If you remain with your original opinion, the chance of seeing the car will be limited to 33.7%. This is the paradox. It seems that there are always two doors left, in which there is one prize, and therefore the probability of winning (change/don’t change) is 50%. But in reality everything is completely different. If the presenter immediately opened the door with the goat, and then asked the player to choose one of two doors, then the chance would really be 50%. But first the player makes his choice and the probability of winning the initially chosen door is 1/3.

  If this choice is repeated many times, its probability will always remain at the level of 1/3, regardless of any further actions of the leader or the player himself. Accordingly, for the two remaining doors, there will always be a probability of 2/3. And because the leader of these two doors always leaves one, then it takes on the value of this probability of 2/3.

  So it turns out The player's initial choice will lead to a win in a third of all cases, and a change of decision will lead to a win in two thirds. That is why this task is called a paradox, because it defies logic and common sense. The human brain is accustomed to working in a pattern, which is why optical illusions, illusions, and paradoxes occur. This is nothing more than a person’s ignorance of a specific issue. Even the logical explanation of the problem written above is not accepted by everyone, and one has to use a more accessible method of enlightenment.

  

  Let's present this problem in a slightly different, more expanded format. There are not 3, but 10 doors, but the conditions are still the same - the player chooses one door, and the presenter opens all the doors and leaves one again. The host can only open doors with a goat. Those. the player again faces a choice - door with a goat / door with a car. Here the conditions are more understandable for the average person to understand.

  It is clear that initially it is very difficult to choose a door with a prize, or rather the probability is 1/10. And it is logical that most likely the car will be behind the remaining of the 9 doors. And because the presenter opens only non-winning ones, then the door that remains open after the presenter and will be offered to the player will be the door with the prize. If such a formulation has caused difficulties for a person, then the conditions can be simplified even more until, as they say, it gets it. This is not a sign of a person’s great or small intelligence, rather it is an excellent test of the subject “are you a humanist or a techie.” Options with two, ten, thousand, etc. doors are identical in essence, but differ in difficulty of perception. The fewer doors, the easier it is to confuse a person.

  

  The appearance of the Monty Hall paradox on sites dedicated to various strategies is more pleasing than sad, especially for bookmakers. The truth is that for now the significance of the Monty Hall paradox is given exclusively to practical purposes. This is more like a clear example that not everything you see is what really is. That the same bookmaker odds may contain not only the real distribution of forces based on statistics and current news from the teams. Players can also move the line without any objective reasons. Here the usual herd reflex () and agreements can take place. Yes, just one big bet on an uncongested event can move the line.

  Although there are also unique people who claim that this paradox can easily be applied to sports betting. Unfortunately, these are statements without any evidence. Let's imagine the Monty Hall paradox in the context of sports betting. To start you need to find an event with equal three chances of success. There are such things, although they are rare. There is a line on football where the odds for one team to win, a draw and the other team are 2.7 - an even line to the point of impossibility. We need to choose our option. Then it is required that at a certain stage one event disappears, and two, the most probable, remain. Until the end of the match, not a single event can be dismissed, even if it is unlikely.

  Over the long haul, it will definitely play out and give its skew to the statistics. But, even if you imagine that it won’t play, then at the stage when there are two options left, these options will already have values ​​commensurate with the original ones. And all because the bookmaker moves the odds during the match. Roughly speaking, when you have to choose between two doors, it will no longer be a goat and a car, but a goat and a bicycle. A goat is zero, a loss is not going anywhere. And the car will turn from a coefficient of 2.7 into a bicycle with a much lower coefficient.

  As a result, although changing the initial decision may increase the winning percentage, the winning itself will have a completely different value. Those. in Monty Hall's paradox the initial conditions do not change, but in sports betting they do. Hence its inapplicability in the fight against bookmakers. On the other hand, who knows? Maybe there is some kind of paradox here, it’s just that no one sees it yet.

  Conclusion

  We continue to strongly recommend using. Leave high-risk financial strategies for casinos or practice gaming accounts. For stable earnings on bets you need the right one, not all kinds of variations. HOW place a bet without understanding FOR WHAT.

  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 the probability of choosing correctly will be 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

  Probability theory is a branch of mathematics that is ready to confuse mathematicians themselves. Unlike the other, precise and unshakable dogmas of this science, this area is teeming with oddities and inaccuracies. A new paragraph, so to speak, was recently added to this section - the Monty Hall paradox. This is, in general, a task, but it is solved in a completely different way from the usual school or university ones.

  Origin story

  People have been racking their brains over the Monty Hall paradox since back in 1975. But it’s worth starting in 1963. It was then that a TV show called Let’s make a deal was released on the screens, which translates as “Let’s make a deal.” Its host was none other than Monty Hall, who presented viewers with sometimes unsolvable problems. One of the most striking became the one he presented in 1975. The problem became part of the mathematical theory of probability and the paradoxes that fit within its framework. It is also worth noting that this phenomenon has caused much debate and harsh criticism from scientists. Monty Hall's Paradox was published in the journal Parade in 1990, and since then has become an even more discussed and controversial issue of all times and peoples... Well, now we move directly to its formulation and interpretation.

  Problem Statement

  There are many interpretations of this paradox, but we decided to present you with the classic one, which was shown in the program itself. So, there are three doors in front of you. Behind one of them there is a car, behind the other two there is one goat each. The presenter invites you to choose one of the doors, and let’s say you stop at number 1. So far, you don’t know what’s behind this very first door, since they open the third one and show you that there’s a goat behind it. Therefore, you have not lost yet, because you have not chosen the door that hides the losing option. Therefore, your chances of getting a car increase.

  

  But then the presenter invites you to change your decision. There are already two doors in front of you, behind one is a goat, behind the other is the desired prize. This is precisely the crux of the problem. It seems that whichever door you choose, the chances are 50/50. But in fact, if you change your mind, you are more likely to win. How so?

  The first choice you make in this game is random. You can’t even remotely guess which of the three doors the prize is hidden behind, so you randomly point to the first one you come across. The presenter, in turn, knows where everything is. He has a door with a prize, a door that you pointed to, and a third without a prize, which he opens for you as the first clue. The second clue lies in his very proposal to change the choice.

  

  Now you will no longer choose one of the three at random, and you can even change your decision to get the desired prize. It is the presenter’s proposal that gives the person the belief that the car is really not behind the door he chose, but behind another one. This is the whole essence of the paradox, since, in fact, you still have to choose (albeit from two, and not from three) at random, but the chances of winning increase. As statistics show, out of 30 players who changed their decision, 18 won the car. And this is 60%. And of the same 30 people who did not change their decision - only 11, that is, 36%.

  Interpretation in numbers

  Now let's give the Monty Hall paradox a more precise definition. The player's first choice splits the doors into two groups. The probability that the prize is located behind the door you chose is 1/3, and 2/3 behind the doors that remain. The leader then opens one of the doors of the second group. Thus, he transfers the entire remaining probability, 2/3, to the one door that you did not choose and that he did not open. It is logical that after such calculations it will be more profitable to change your decision. But it is important to remember that there is still a chance to lose. Sometimes the presenters are disingenuous, since you can initially point at the right prize door, and then voluntarily refuse it.

  We are all accustomed to the fact that mathematics, as an exact science, goes hand in hand with common sense. What matters here are numbers, not words, precise formulas, not vague reflections, coordinates, not relative data. But its new section, called probability theory, blew up the entire familiar pattern. Problems in this area, it seems to us, do not fall within the framework of common sense and completely contradict all formulas and calculations. We suggest below that you familiarize yourself with other paradoxes of probability theory that have something in common with the one described above.

  Boy and girl paradox

  The problem, at first glance, is absurd, but it strictly obeys the mathematical formula and has two solutions. So, a certain man has two children. One of them is probably a boy. What is the probability that the second one will be a boy?

  Option 1. We consider all combinations of two children in a family:

  • Girl/girl.
  • Girl boy.
  • Boy/girl.
  • Boy/boy.

  The first combination obviously does not suit us, therefore, based on the last three, we get a 1/3 probability that the second child will be a small man.

  

  Option 2. If we imagine such a case in practice, discarding fractions and formulas, then, based on the fact that there are only two sexes on Earth, the probability that the second child will be a boy is 1/2.

  This experience shows us how cleverly statistics can be manipulated. So, the “sleeping beauty” is injected with sleeping pills and given a coin. If it lands on heads, she is woken up and the experiment ends. If it lands on heads, they wake her up, immediately giving her a second injection, and she forgets that she woke up, and after that they wake her up again only on the second day. After fully awakening, the “beauty” does not know on what day she opened her eyes, or what the probability is that the coin landed on heads. According to the first solution, the probability of landing heads (or tails) is 1/2. The essence of the second option is that if the experiment is carried out 1000 times, then in the case of heads the “beauty” will be awakened 500 times, and with rare ones - 1000. Now the probability of getting tails is 2/3.