Monty Hall problem

The Monty Hall problem, three -door problem, Monty Hall problem or the Monty Hall dilemma is a task with respect to probability theory. The task is loose the game show Let's Make a Deal modeled, which in the German language in the variant go the whole hog! was known. The names refer to Monty Hall, the host of Let's Make a Deal, or the goats win in a known problem formulation beside the right price, a car, as mockery prices.

The Monty Hall problem is often used as an example of how the human mind to fallacies tends, when it comes to determining probabilities, and is the subject of a prolonged public debate.

The task itself goes back to the biostatistician Steve Selvin, who published them as a letter to the editor in the American Statistician in 1975. Worldwide known and the subject of a controversial debate, the problem was though. The local version is based only by its publication in Marilyn vos Savant's "Ask Marilyn " column of Parade magazine in 1990 to a reader's letter, the vos Savant by Craig F. Whitaker had from Columbia, Maryland, received.

" Suppose you were in a game show and had the choice between three gates. Behind one of the doors is a car, behind the others are goats. Select a goal, say, door number 1, and the show host, who knows what is behind the doors, opens another door, say, number 3, behind which stands a goat. He will now ask you: you want the gate number 2 ' Is it advantageous to change the choice of the gate "?

The question in this form is under-determined, the correct answer depends on what additional assumptions are made. Vos Savant gave the answer " Yes, you should switch. The first selected gate has the chance of winning 1/3, but the second goal has a chance of winning 2/3 ". Vos Savant's answer is, although under additional assumptions correct, even under these additional assumptions for many people very counter intuitive. As a result vos Savant achieved numerous letters, according to information provided tens of thousands, which mainly questioned the correctness of their response.

The Monty Hall problem is the subject of ongoing public debate and scientific investigation.

• 4.1 The balanced Moderator 4.1.1 Simple Explanation
• 4.1.2 Tabular solution
• 4.1.3 Formal mathematical solution

• 4.2.1 Tabular solution
• 4.2.2 Formal mathematical solution

• 6.1 Notes on the Literature
• 6.2 The question is qualitative not quantitative
• 6.3 frequentist view
• 6.4 The influence of the moderator behavior
• 6.5 Bayesian (or Bayesian ) Visual
• 6.6 Comparison of different solutions 6.6.1 Numbering the gates
• 6.6.2 Empirical examination of a related to the Moderator solution behavior
• 6.6.3 decision situations with different odds
• 6.6.4 The two a 2/3: 1/3-Vorteil be forecast solutions

• 6.7.1 moderator can also open the doors with the car
• 6.7.2 moderator can also open the selected first goal
• 6.7.3 Game Theoretical Approach

The experiential response

If one asks the question people who had not yet dealt with the problem, this often suggests that the odds for Goals 1 and 2 are equal. The reason is often stated that you knew nothing about the motivation of the game show host, port 3 with a goat behind it to open and offer a change. It therefore grab the principle of indifference.

The intuition in understanding the letter to the editor assumes that it is the description of a one- game situation in the problem. In addition, the response of a certain familiarity with game testifies shows like Go the whole hog, where the show Master ( Moderator) plays an active role and unpredictable. In contrast to the problem variations in which the moderator is reduced to a bound to fixed rules of conduct " stooge ", may realistically be assumed that in his decisions he is completely free ( Monty Hall: "I am the master of the house !"). This freedom can be illustrated with some examples, before each game car and goats were redistributed randomly behind the three gates. Because the candidates this game show for which they have applied to be participants know, give them the unpredictability of the moderator is of course aware of.

Given the different behavior options of the moderator Doris should consider carefully their chances of winning. If she believes that the moderator was nice to her and she wants to be dissuaded from their first wrong choice, then they should change. However, when it says that you, the moderator was not well disposed and she wants to divert attention from their first, right choice, then they should stay at Gate 1. When Doris the moderator can not judge - in letter no relevant warnings are given - she still has the ability to invoke the principle of indifference and assess their chances of winning through rates than 1/2. The answer to the question "Is it advantageous to change the choice of the gate " is in their case, " Not necessarily. "

Although the question of the letter to the editor is already been answered, a proposal was made to support Doris in their decision and to give it a real 50-50 chance of winning. For this purpose, it is assumed that it has the opportunity to decide on the toss of a fair coin for one of the two remaining doors. In this way they can ensure that their probability of winning is independent of the intentions of the moderator exactly 1/2.

Marilyn vos Savant Reply

By Marilyn vos Savant Reply to the letter to the problem generated internationally well outside the world of high attention and led to considerable controversy. Her answer was:

" Yes, you should switch. The first selected gate has the chance of winning 1/3, but the second goal has a chance of winning 2/3. Here is a good way to think of the action. Suppose there were 1 million gates and select gate number 1 then opens the moderator, who knows what is behind the gates, and always avoids a gate with the price, all doors up to gate number 777,777th you but would immediately switch to this gate, or not? "

Marilyn vos Savant is not taking into account a specific motivation of the presenter; there is aloud letter by no means impossible that the moderator opens the only reason why a Ziegentor to distract the candidate from his first successful election. Instead vos Savant summarizes the letter obviously so on that game show runs again according to the same pattern:

Thus it receives as a solution, the average winning percentage of all possible combinations of gates, who are elected by the respective candidates and can be opened by the moderator then. Since the first election of a candidate as any, and the distribution of car and goats behind the gates is considered random, each of the nine options may be equally likely considered as:

Win three out of nine candidates, if they stay with their first choice, get while six of the nine candidates by changing the car. A candidate has change so an average chance of winning p = 2/3.

This solution may also illustrated graphically werden.In the images of the following table is the chosen target arbitrarily shown as the left gate:

Strategic solution

Because of the concept of vos Savant and taking into account of its proposed change strategy can be an alternative view of the expiry of the game show formulate:

For example, would leave open gate 2 and gate 3 a candidate. So he chooses door 1, which remains sealed, and then switches to port 2 when the moderator gate has 3 open, or vice versa. The candidate has so obvious an average chance of winning p = 2/3. Accordingly, it would be for a candidate who repeatedly likely to participate in this game show, beneficial, the choice of the gate to change forever.

Controversies

There are mainly the following main arguments that lead to doubt of vos Savant's answer. While the first argument is without merit and is based on wrong applied probability theory, illustrate the other arguments that the original problem allows a variety of interpretations:

• Under the assumption that the Showmaster follow the game rules explained in the next section, is a change of the gate is not bad. The odds of winning for the second goal was never 2/3, but generally only 1/2, because after opening a door with a goat behind it, only two closed gates stood for selection. The chances are therefore always equally distributed on both goals.
• The question in the letter does not contain any evidence that the Showmaster follows a certain rule of conduct. Such a rule could be only under the assumption deduce that the game would be repeated several times under the same conditions: select any gate, the show host opens another door, behind which a goat is, and you may change the choice of door. From such a repetition of the game but there is no mention in the letter. So based vos Savant's answer to the additional assumptions that result in this form not necessarily from the letter to the editor.
• Marilyn vos Savant's interpretation does not apply to the gates specifically named in the question, and thus they can possibly existing preferences of the moderator with respect to individual goals in mind. Therefore it receives as win probability 2/3 by changing that is not valid with any moderator behavior. Accordingly, the above table, which only illustrates the average probabilities of such preferences is not starting correctly.

The first argument is refuted by the balanced moderator, the second is performed on the basis of experience-based answer and the third based on the lazy moderator.

The Monty Hall Standard problem

Because formulated in the letter to the editor of Whitaker task some scientists not uniquely solvable appeared, a reformulation of the goat problem was proposed by them. That person, the Monty Hall Standard problem reformulation that such should lead by Marilyn vos Savant the same solution, provides certain additional information available that make the experiential response is invalid and considered in contrast to the interpretation of vos Savant also the specific game situation:

"Suppose you are in a game show and have the choice of three doors. Behind one door is a car that is behind the other one goat. The car and the goats were randomly distributed on the doors before the show, and you have no information about the position of the car. The rules are: Once you have selected a goal, this is initially closed. The show host Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors. Behind the open door of it must be located a goat. After Monty Hall has opened a door with a goat, he will ask you if you stay with your first choice or want to switch to the last remaining door. Suppose you choose door 1, and the show host opens door 3 with a goat. He then asks: ' Would you like to switch to port 2 ?'. Is it advantageous to change your choice? "

In particular, the moderator has the ability to be free to decide which goal he opens when he can choose between two goats gates has ( So you have first selected the Auto- gate). Divided into individual steps so results in the following rules, which are the candidate who can win a car, known:

Significance of the additional assumption on the behavior of the moderator:

With such an additional assumption in each case arises another problem that can result in different chances of winning at the Door Selection of the candidate. To this end, it is always assumed that the candidate knows the moderator imputed decision procedure.

The balanced Moderator

For this solution, the following additional assumption is made:

• Case A: If the candidate has chosen the gate with the car, the moderator opens randomly selected with equal probability one of the other two doors, behind which there is always a goat.

How to choose the penultimate step in initially chosen door 1 and the moderator then gate 3 is opened with a goat behind the candidate?

Simple Explanation

The car is with probability 1/3 behind the goal initially chosen by the candidates 1 Because of the symmetry in the rules, in particular because of rules 4 and 5, this probability is not affected by the opening of another door with a goat behind it. Therefore, after opening the door 3 is the car with 2/3-Wahrscheinlichkeit behind Gate 2, and a change leads with probability 2/3 to success.

Tabular solution

For the explanation, it is assumed that the candidate has chosen to start Tor 1 and then umentscheidet. For situations in which the candidate has 2 or 3 is selected, the moderator opens the gates and other gates accordingly, applies a similar representation. There must be six cases are considered in order to model the equal probability of opening the gates 2 and 3 by the moderator under Rule 4 can. Corresponding to a random experiment in which the two can be distinguished from each other, goats, and any distribution of the car and goats behind the three ports is equally likely ( Laplace experiment).

To evaluate the table, the cases must now be considered in which the moderator opens the door 3 (which is the condition). These are the cases 2, 4 and 5 can be seen that by changing wins in two out of three of these cases the candidate. Under the conditions that the candidate has initially chosen door 1 and the host opens door 3 with a goat behind it, the car is so in 2/3 of the cases behind gate 2 addition can be easily read from the table that when the moderator instead of gate 3, gate 2 opens, the contestant wins the car by changing also in two of three cases. The candidate should therefore change its choice in favor of gate 2.

Formal mathematical solution

There are the events defined:

It is the following situation: The candidate has chosen Door 1, and the moderator has then opened the gate 3. Is it worth it to go for the candidate? What is the probability that the car is behind door 2? So Wanted is the conditional probability that the car is behind door 2, when it is known that it is not behind door 3. You can determine this probability with Bayes' Theorem.

On the basis of the task (rules 1, 4 and 5 ), the following requirements apply:

The application of the theorem of Bayes results in:

The candidate should change so to double his chances of winning from the beginning 1/3 to now 2/3.

Lazy Moderator

For this solution, the following additional assumption is made:

• Case A: The moderator who does not like travels over large distances, opens prefer Goal 3 because he has near his location as master of ceremonies there. So when behind the gate one chosen by the candidate would be the car, then he would open with security gate 3, but under no circumstances Gate 2

Tabular solution

For the following explanation, it is assumed that the candidate has chosen to start Tor 1. For situations in which the candidate has 2 or 3 is selected, the moderator opens the gates and other gates accordingly, applies a similar representation. Although it would suffice here to consider the first three game situations, six cases are distinguished in order to model the problem similar to the above tabular solution when balanced moderator can. Each game situation is thus considered twice. Corresponding to a random experiment in which the two can be distinguished from each other, goats, and any distribution of the car and goats behind the three ports is equally likely ( Laplace experiment).

To evaluate the table, the cases must now be considered in which the moderator opens the door 3 (which is the condition). These are the cases 1, 2, 4 and 5 can be seen that gains by changing only in two out of four of these cases the candidate. His winning percentage is thus only p = 1/2. It can also be easily read from the table that when the moderator Gate 2 opens, the candidate sure to win, if he goes to port 3.

Formal mathematical solution

It is the following situation: The candidate has chosen Door 1, and the moderator has then opened the gate 3. Then apply the following mathematical relations in the light of the above defined sets of events:

The application of the theorem of Bayes then gives for the conditional probability that the car is behind door 2:

For the conditional probability that the car is actually located behind Gate 1, but also applies

The gain behind Gate 2 is just as likely as the profit behind gate 1 The candidate can therefore remain as good at Gate 1 in this case as switch to port 2. Has opened the moderator Tor 3, his chance of winning is independent of the decision 1/2.

The unbalanced Moderator

In this solution, it is assumed that the following additional assumption:

• Case A: If the moderator has the option behind choose from two goals each with a goat a gate ( the candidate, therefore, the door with the car behind it selected), and then he opens the door with the highest possible number with the probability and the gate with the lower number with the probability.

Then the following mathematical relationships apply, taking into account the above-defined sets of events:

The application of the theorem of Bayes then gives for the conditional probability that the car is behind door 2:

This calculation describes the general case can be derived from the " lazy moderator " () as special cases of the " balanced moderator " () and the.

The general solution

From the consideration of unbalanced moderator can be concluded that, regardless of its respective preference for a particular Ziegentor the probability of winning is always at least 1/2, is by changing after you open a goat door, on average, even 2/3. As long as the moderator is forced in accordance with the rules of the game, always to open an unelected Ziegentor and then offer an exchange, a candidate should change his choice of the gate so in every case.

The older Monty Hall Problem

In February 1975, the academic journal The American Statistician published a letter by Steve Selvin, then an assistant professor of biostatistics at the University of California at Berkeley, to the editor. In this letter, entitled " A Problem in Probability ," he suggested a word problem as an exercise in probability theory before. The given solution he resembles the table, as shown in the section on vos Savant's answer. In August of the same year another letter from the same author published with the title "On the Monty Hall problem ," in which he referred to his first letter and responded to objections from a reader regarding his proposed solution. So at this stage appeared for the first time the term " Monty Hall problem " in the media space on. In his second letter Selvin presented further arguments in favor of its solution, including a formal mathematical calculation using conditional probabilities. He added that his calculations on specific, non-explicit, based assumptions that affect the behavior of the moderator Monty Hall. He also quoted a reader who pointed out that the critical assumptions regarding the moderator behavior are necessary in order to solve the problem at all, and that the initial distribution is only part of the problem posed, while here but a subjective decision problem acted. It is natural to see this early Monty Hall problem as a precursor of what is now known as the Monty Hall problem question, including the dispute over the then already disputed the additional assumptions regarding the conduct of the moderator.

Overview of the literature on " the " Monty Hall Problem

Notes on the literature

In the publications on the Monty Hall problem ( Monty Hall problem) partially different problems are examined, sometimes even different models in a publication. Among others is noteworthy that authors such as Gill put their solution vos Savant's original basis and make their additional assumptions in the course of their analysis explicitly. Other authors such as Lucas, however, use a problem formulation which requires the moderator from the outset certain rules of behavior.

The question is qualitative not quantitative

In relation to the different solutions, as they were also reproduced above, summarizes idol " CHANGE IS WORSE THAN NEVER STAY! " ( Uppercase according to reference ). Had on this issue as early as 1991 Morgan et al., Which made ​​the "discoverer " of the additive based on assumptions about the behavior of solutions moderator attention. Despite this qualitative agreement, and the fact that the problem "Is it advantageous to change the choice of the gate? " Asks for an action and not according to a probability, the subject of intense discussion are the assumptions that lead to different probability values ​​, again and again. So only the bibliography of published in 2009 book The Monty Hall Problem of Rose House contains over one hundred publications.

Frequentist view

After Georgii inter alia, by Marilyn vos Savant answer given based on the fact that other words " implicitly assumes the frequentist interpretation of conditional probabilities [ is ], which presupposes the repeatability of the process and therefore hard and fast rules. ": Will the game show after this previously held fixed rules, wins a candidate who keeps changing strategy, with twice the chance of winning against a candidate who pursues the non- exchange strategy. This chance advantage applies even if the candidate only once participates, and indeed for every goal initially selected. Empirically can the advantage - in terms of quantity - check if a series of tests will be organized, which is statistically evaluated.

Specifically, this form of a frequentist solution is based on the number of textbook writers, partly in the context of a representation of multiple solutions. The winning probability of 2/ 3 for a Torwechsel refers to the time prior to opening a gate by the moderator.

Influence the behavior moderator

The frequentist approaches noted Georgii in his textbook critical that what matters is " how the player evaluates the behavior of the moderator ". Therefore, a subjective interpretation is more appropriate. However, most textbook authors forego the Berücksichtiung such a subjective assessment of the moderator behavior. Specifically, they assume that the moderator Outsmart balanced, that is, that he makes the selection of the gate according to a uniform distribution. Thus, this approach is the most frequent are present in the literature explain why a Torwechsel with the probability of 2/3 leads to profits. This winning probability of 2/ 3 for a Torwechsel refers to the time after opening a gate by the moderator.

Studies in which the candidate is the moderator also evaluates the effect of selecting its Skinned not equally likely to have been the first time in 1991 by Morgan et al. and independently published in 1992 by Gillman.

Bayesian (or Bayesian ) Visual

The fact that "this" is absolutely different results for Monty Hall problem, is after Georgii apart from the different interpretable rules of the game show as to whether the event, which is referred to the question, "is part of a fixed rule of the game or not. " Philosophical uncertainty about the importance of conditional probabilities come here aggravated by. In the same way, but in more detail, Götz refers to two " different probability concepts underlying the respective viewpoints based. " The classic solution is to check frequentist and empirical. In contrast, deliver the " Bayesian solution ... a valuation basis of a single situation. How to behave the candidate here and now, after the GM has opened a door? ... One wonders, after state probabilities or finding probabilities (and not according to probabilities of future random events) " In other words. The candidate makes after the door opening by the moderator evaluating its two courses of action depending on which basic behavior he reports to the moderator. Here, the extreme case of a lazy moderator is characterized by the answer to the following question: "If the moderator, after seeing my decision for a goal, selected the he just opened gate also in all other circumstances, provided that he is only possible - would have been " - no car behind it?

Bayesian tests were first described by Morgan et al. performed, on the basis of their results, in which the presenter for an assumed a priori probabilities selects the gate to open at random according to.

Comparison of the different solutions

Numbering the gates

The last section made ​​characterization of the behavior of a lazy moderator shows that a solution in this regard is not bound to a numbering of the gates (usually " candidate chooses goal 1st master opens door 3, whenever it is possible ").

Empirical examination of a related to the Moderator solution behavior

For example, to found for the variant of a lazy moderator 1/2: 1/2-Lösung be tested empirically, so it should be remembered that the derived based on this statement refers to a conditional event. In a test set of 300 Game Shows, which are carried out in accordance with the additional assumption lazy moderator, so do not go through about 100 shows the event that is the subject of the investigation. Specific reason for this is that with a hidden rear gate 3 car of the moderator is forced to open Gate 2. Such game histories are but outside the study area, so that the gains achieved by a Torwechsel always have to be excluded from the test series analysis.

Decision-making situations with different odds

The "global" set of all possible decision situations Torwechsel strategy brings an overall 2/3-zu-1/3-Vorteil. However, by an asymmetric gameplay decision situations arise in which a Torwechsel compared with the average is promising or less promising. Such effects are obvious in terms of an asymmetric probability distribution at the draw of Gewinntors, but they can, as the results for the lazy moderator show also be caused by an asymmetric behavior moderator. When moderator behavior, however, the possible deviations for the probability of winning the Torwechsel from the a priori value 2/3 down are limited, since the value of 1/2 can not be reached, because " change is never worse than Stay " - see above.

Both a 2/3: 1/3-Vorteil be forecast solutions

Although the "classic" vos Savant - solution consistent with the solution for the balanced moderator for a Torwechsel a 2/3: predicts 1/3-Vorteil, their viewing angle and arguments are very different: One is an a priori probability of the situation immediately before the decision of the moderator specified for an openable gate. The other time refers to the probability of the date when the moderator "his" door has been opened, however, the additional assumption is made that the moderator has made his choices equally likely. The fact that the two approaches provide the same probability of winning follows, from a consideration of symmetry, deriving the a-posteriori value from the a priori value.

More mathematically investigated variants

In addition to the described above interpretation "of the" goat problem, there are other variants which have been studied in the literature. In general, it should be noted to the fact that the authors - there is no consensus that mathematical model, " the " Monty Hall problem and its corresponding question - as in regard to the interpretations presented above. Some of the models are used only for the purpose of illustrative comparison:

Moderator can also open the doors with the car

Lucus, Rose House, Madison and Schepler analyze, among other things, the variant in which the presenter his goal randomly selects among the two remaining gates and where appropriate also opens the door to the car. A short calculation confirms the intuitively obvious assumption that for this variant in the event that a gate is opened with goat, the probability of winning is when replacing 1/2.

Moderator can also open the selected first goal

Georgii can be in one of the two investigated variants of it also the case that the moderator opens the first player selected by the door with a goat. When the moderator chooses this randomly with equal probability between the two goat doors, the probability of winning for a change according to the " response of the critics " is also 1/2 when he opens an unelected Ziegentor.

Game Theoretical Approach

The fact that the behavior of the moderator and the assessment of this behavior by the candidate, the chances of winning at a Torwechsel was taken by some authors as a starting point game theoretical studies of the goat problem. The additional assumption about the probability of the facilitator is seen as a mixed strategy in terms of a two-person game that even has a zero-sum character. Also included in the sequential game play is also hiding the car, which is seen as a first train of the moderator. With a simple argument that uses obvious, symmetrical with respect to the gates strategies for both players, Gill was able to show that the minimax value 2 /3.

The amount of minimax strategies for both players was determined by Gnedin. In this case, the candidate has only one minimax strategy in which he auslost his first goal elected according to a uniform distribution and then always changes the target. The statement is remarkable, since it requires no a priori assumption about the behavior of the moderator and it still makes statements for each individual appearing in-game decision-making situation. An even stronger argument for the candidate who never maintain the initially chosen goal, results from Gnedins dominance analyzes for strategies.