# Monty Hall

Edited by Charles B. Cross (University of Georgia)
 Summary The Monty Hall Problem is a puzzle derived from the game show Let's Make a Deal, which first aired on American television during the 1960's and was for many years hosted by Monty Hall. Unlike most other philosophically interesting decision problems, the Monty Hall Problem has a straightforwardly correct solution, but this solution is easy to miss. The game show contestant is shown a series of closed doors and told that s/he may have what is behind exactly one of Door #1, Door #2, or Door #3. S/he is told that there is a new car behind exactly one of the three doors and nothing of much value behind either of the other two. The contestant selects exactly one of the three doors (say, Door #1), but this door is not opened. At this point the host opens one of the other doors (say, Door #2) revealing that the car is not behind that door. The host now asks whether the contestant would like to keep what is behind Door #1 or switch and take what is behind the unopened Door #3. Assuming that (1) the host must open a door that is not the contestant's initial choice and that does not have a car behind it, and (2) if the host has two non-car-concealing doors to choose from, the host will choose at random which of them to open, and (3) each door has an initial probability of 1/3 of concealing the car, it turns out that when the host opens Door #2, the probability of finding the car behind Door #3 increases from 1/3 to 2/3, which makes switching to Door #3 the correct decision. This result is surprising and counterintuitive, since, until the assumptions of the problem are taken into account, it may seem that the host has simply eliminated Door #2 and thereby given each of the remaining two doors a probability of 1/2 of concealing the car.
 Key works A version of the Monty Hall Problem appears in Steve Selvin, A Problem in Probability (Letter to the Editor), The American Statistician 29, No. 1 (Feb., 1975), p. 67. The problem was popularized in Marilyn vos Savant's column, Ask Marilyn, in Parade Magazine (9 September 1990, p. 16; 2 December 1990, p. 25; 17 February 1991, p. 12; 7 July 1991, p. 26). Since there is no issue concerning which solution to the Monty Hall Problem is correct, discussion of the problem in the philosophical literature has focused on how the problem is related to other issues, such as single case probabilities (Baumann 2005, Baumann 2008, Sprenger 2010), self-locating beliefs (Bradley 2007), and other decision problems including Judy Benjamin, Sleeping Beauty, and Doomsday (Bovens & Ferreira 2010, Bradley & Fitelson 2003).
 Introductions For a presentation of the Bayes Theorem calculations that yield the correct solution, see Cross 2000.
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1. Jean Baratgin & Guy Politzer (2011). Updating: A Psychologically Basic Situation of Probability Revision. Thinking and Reasoning 16 (4):253-287.
The Bayesian model has been used in psychology as the standard reference for the study of probability revision. In the first part of this paper we show that this traditional choice restricts the scope of the experimental investigation of revision to a stable universe. This is the case of a situation that, technically, is known as focusing. We argue that it is essential for a better understanding of human probability revision to consider another situation called updating (Katsuno & Mendelzon, 1992), (...)

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2. Peter Baumann (2008). Single-Case Probabilities and the Case of Monty Hall: Levy's View. Synthese 162 (2):265 - 273.
In Baumann (American Philosophical Quarterly 42: 71–79, 2005) I argued that reflections on a variation of the Monty Hall problem throws a very general skeptical light on the idea of single-case probabilities. Levy (Synthese, forthcoming, 2007) puts forward some interesting objections which I answer here.

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3. Peter Baumann (2005). Three Doors, Two Players, and Single-Case Probabilities. American Philosophical Quarterly 42 (1):71 - 79.
The well known Monty Hall-problem has a clear solution if one deals with a long enough series of individual games. However, the situation is different if one switches to probabilities in a single case. This paper presents an argument for Monty Hall situations with two players (not just one, as is usual). It leads to a quite general conclusion: One cannot apply probabilistic considerations (for or against any of the strategies) to isolated single cases. If one does that, one cannot (...)

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4. Consider van Fraassen's ( 1981) Judy Benjamin (JB) problem. Judy is dropped in an area that is divided vertically in Blue (B) and Red (R) and horizontally in Headquarters (Q) and Second Company (S). These divisions define four quadrants, as in Figure 1 (roman script headings). Judy initially believes that there is an equal chance of being in each quadrant. She is then told by a fully reliable source that if she is in R, then there is a chance of (...)

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5. Darren Bradley (2007). Bayesianism And Self-Locating Beliefs. Dissertation, Stanford University
How should we update our beliefs when we learn new evidence? Bayesian confirmation theory provides a widely accepted and well understood answer – we should conditionalize. But this theory has a problem with self-locating beliefs, beliefs that tell you where you are in the world, as opposed to what the world is like. To see the problem, consider your current belief that it is January. You might be absolutely, 100%, sure that it is January. But you will soon believe it (...)

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6. Darren Bradley & Branden Fitelson (2003). Monty Hall, Doomsday and Confirmation. Analysis 63 (277):23–31.
We give an analysis of the Monty Hall problem purely in terms of confirmation, without making any lottery assumptions about priors. Along the way, we show the Monty Hall problem is structurally identical to the Doomsday Argument.

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7. Darren Bradley & Branden Fitelson (2003). Monty Hall, Doomsday and Confirmation. Analysis 63 (1):23-31.

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8. Charles B. Cross (2000). A Characterization of Imaging in Terms of Popper Functions. Philosophy of Science 67 (2):316-338.
Despite the results of David Lewis, Peter Gärdenfors, and others, showing that imaging and classical conditionalization coincide only in the most trivial probabilistic models of belief revision, it turns out that imaging on a proposition A can always be described via Popper function conditionalization on a proposition that entails A. This result generalizes to any method of belief revision meeting certain minimal requirements. The proof is illustrated by an application of imaging in the context of the Monty Hall Problem.

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9. Joseph Ellin, Monty Hall No Newcomb Problem. Proceedings of the Heraclitean Society 17.
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10. Arthur Falk (1993). Summer 1991: The "Monty Hall" Problem; Fall 1993: The Two Envelopes Puzzle; And Now: Doomsday. Proceedings of the Heraclitean Society 17:64.
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12. Branden Fitelson (2007). Likelihoodism, Bayesianism, and Relational Confirmation. Synthese 156 (3):473 - 489.
Likelihoodists and Bayesians seem to have a fundamental disagreement about the proper probabilistic explication of relational (or contrastive) conceptions of evidential support (or confirmation). In this paper, I will survey some recent arguments and results in this area, with an eye toward pinpointing the nexus of the dispute. This will lead, first, to an important shift in the way the debate has been couched, and, second, to an alternative explication of relational support, which is in some sense a "middle way" (...)

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13. This research examined choice behaviour and probability judgement in a counterintuitive reasoning problem called the Monty Hall problem (MHP). In Experiments 1 and 2 we examined whether learning from a simulated card game similar to the MHP affected how people solved the MHP. Results indicated that the experience with the card game affected participants' choice behaviour, in that participants selected to switch in the MHP. However, it did not affect their understanding of the objective probabilities. This suggests that there is (...)

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14. The Monty Hall Problem (MHP), a process of two-stage decision making, was presented in atypical form via a custom software game. Differing from the normal three-box MHP, the game added one additional box on-screen for each game—culminating on game 23 with 25 on-screen boxes to initially choose from. A total of 108 participants played 23 games (trials) in one of four conditions; (1) “Vanish” condition—all non-winning boxes totally removed from the screen; (2) “Empty” condition—all non-winning boxes remain on-screen, but with (...)

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15. Stefan Krauss & X. T. Wang (2003). The Psychology of the Monty Hall Problem: Discovering Psychological Mechanisms for Solving a Tenacious Brain Teaser. Journal of Experimental Psychology: General 132 (1):3.

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16. Ken Levy (2007). Baumann on the Monty Hall Problem and Single-Case Probabilities. Synthese 158 (1):139-151.
Peter Baumann uses the Monty Hall game to demonstrate that probabilities cannot be meaningfully applied to individual games. Baumann draws from this first conclusion a second: in a single game, it is not necessarily rational to switch from the door that I have initially chosen to the door that Monty Hall did not open. After challenging Baumann's particular arguments for these conclusions, I argue that there is a deeper problem with his position: it rests on the false assumption that what (...)

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17. Ken Levy (2007). Baumann on the Monty Hall Problem and Single-Case Probabilities. Synthese 158 (1):139 - 151.
Peter Baumann uses the Monty Hall game to demonstrate that probabilities cannot be meaningfully applied to individual games. Baumann draws from this first conclusion a second: in a single game, it is not necessarily rational to switch from the door that I have initially chosen to the door that Monty Hall did not open. After challenging Baumann’s particular arguments for these conclusions, I argue that there is a deeper problem with his position: it rests on the false assumption that what (...)

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18. Lore Saenen, Wim Van Dooren & Patrick Onghena (2014). A Randomised Monty Hall Experiment: The Positive Effect of Conditional Frequency Feedback. Thinking and Reasoning 21 (2):176-192.
The Monty Hall dilemma is a notorious probability problem with a counterintuitive solution. There is a strong tendency to stay with the initial choice, despite the fact that switching doubles the probability of winning. The current randomised experiment investigates whether feedback in a series of trials improves behavioural performance on the MHD and increases the level of understanding of the problem. Feedback was either conditional or non-conditional, and was given either in frequency format or in percentage format. Results show that (...)

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19. Brian Skyrms (1966). Choice and Chance. Belmont, Calif.,Dickenson Pub. Co..

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20. Jan Sprenger (2010). Probability, Rational Single-Case Decisions and the Monty Hall Problem. Synthese 174 (3):331 - 340.
The application of probabilistic arguments to rational decisions in a single case is a contentious philosophical issue which arises in various contexts. Some authors (e.g. Horgan, Philos Pap 24:209–222, 1995; Levy, Synthese 158:139–151, 2007) affirm the normative force of probabilistic arguments in single cases while others (Baumann, Am Philos Q 42:71–79, 2005; Synthese 162:265–273, 2008) deny it. I demonstrate that both sides do not give convincing arguments for their case and propose a new account of the relationship between probabilistic reasoning (...)

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21. Jan Sprenger (2010). Probability, Rational Single-Case Decisions and the Monty Hall Problem. Synthese 174 (3):331-340.