1 A player who stays with the initial choice wins in only one out of three of these equally likely possibilities, while a player who switches wins in two out of three. Richard Gill[54] analyzes the likelihood for the host to open door 3 as follows. Following Gill,[56] a strategy of contestant involves two actions: the initial choice of a door and the decision to switch (or to stick) which may depend on both the door initially chosen and the door to which the host offers switching. Several critics of the paper by Morgan et al,[38] whose contributions were published alongside the original paper, criticized the authors for altering vos Savant's wording and misinterpreting her intention. [26] People strongly tend to think probability is evenly distributed across as many unknowns as are present, whether it is or not.[27]. Ambiguities in the Parade version do not explicitly define the protocol of the host. [21] In her book The Power of Logical Thinking,[22] quotes cognitive psychologist Massimo Piattelli Palmarini [it]: "No other statistical puzzle comes so close to fooling all the people all the time [and] even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer." Other possible behaviors than the one described can reveal different additional information, or none at all, and yield different probabilities. [20], The discussion was replayed in other venues (e.g., in Cecil Adams' "The Straight Dope" newspaper column[14]) and reported in major newspapers such as The New York Times.[4]. "You pick door #1. If the host picks randomly q would be 1/2 and switching wins with probability 2/3 regardless of which door the host opens. exciting connections of game theory to other fields such as computer (some but not all of which have solutions in the back of the When only the highest bidder can win, 14.10.3. After the player picks his card, it is already determined whether switching will win the round for the player. Game Theory, Alive by Anna R. Karlin and Yuval Peres . These probabilities can be determined referring to the conditional probability table below, or to an equivalent decision tree as shown to the right. The host must always offer the chance to switch between the originally chosen door and the remaining closed door. After a reader wrote in to correct the mathematics of Adams's analysis, Adams agreed that mathematically he had been wrong. The problem continues to attract the attention of cognitive psychologists. We live in a highly connected world with multiple self-interested direct effects, but also how it influences the incentives of ", Solutions using conditional probability and other solutions, Conditional probability by direct calculation, Similar puzzles in probability and decision theory, "Pedigrees, Prizes, and Prisoners: The Misuse of Conditional Probability", "Partition-Edit-Count: Naive Extensional Reasoning in Judgment of Conditional Probability", Journal of Experimental Psychology: General, Personality and Social Psychology Bulletin, "Are birds smarter than mathematicians? (including some modern ones), clear and well-motivated exposition, a It offers an appealing and versatile selection of topics Stable matching and allocation, 10.2. The simple solutions show in various ways that a contestant who is determined to switch will win the car with probability 2/3, and hence that switching is the winning strategy, if the player has to choose in advance between "always switching", and "always staying". As N grows larger, the advantage decreases and approaches zero. N good selection of examples, and a nicely-chosen selection of exercises Is it to your advantage to switch your choice?[9]. evolutionary stability), and learning theory. Other partisan games played on graphs, 2.4. Twelve such deterministic strategies of the contestant exist. A version of the problem very similar to the one that appeared three years later in Parade was published in 1987 in the Puzzles section of The Journal of Economic Perspectives. This is Among the simple solutions, the "combined doors solution" comes closest to a conditional solution, as we saw in the discussion of approaches using the concept of odds and Bayes theorem. Sperner’s Lemma in higher dimensions*, 6.3.3. MAA Member Price: $67.50. [48][49] In contrast most sources in the field of probability calculate the conditional probabilities that the car is behind door 1 and door 2 are 1/3 and 2/3 given the contestant initially picks door 1 and the host opens door 3. [1][2] The first letter presented the problem in a version close to its presentation in Parade 15 years later. "Game Theory, Alive" is a wonderful book and is to be highly recommended, either for teaching or self-study...this reviewer would not be surprised if "Game Theory, Alive" becomes the standard text for an introductory course on Game Theory. Given that the car is not behind door 1, it is equally likely that it is behind door 2 or 3. Switching wins the car two-thirds of the time. For example, if the host is not required to make the offer to switch the player may suspect the host is malicious and makes the offers more often if the player has initially selected the car. The player initially chooses door i = 1, C = X1 and the host opens door i = 3, C = H3. [14] Adams initially answered, incorrectly, that the chances for the two remaining doors must each be one in two. Vera has to decide whether [10] Out of 228 subjects in one study, only 13% chose to switch. These are the only cases where the host opens door 3, so if the player has picked door 1 and the host opens door 3 the car is twice as likely to be behind door 2. As Keith Devlin says,[15] "By opening his door, Monty is saying to the contestant 'There are two doors you did not choose, and the probability that the prize is behind one of them is 2/3. p If the host chooses uniformly at random between doors hiding a goat (as is the case in the standard interpretation), this probability indeed remains unchanged, but if the host can choose non-randomly between such doors, then the specific door that the host opens reveals additional information. Raquel has to choose whether to pursue training that costs $1;000 to herself or not. Even if the host opens only a single door ( level. opportunities. world-view. Therefore, they are both equal to 1/3. The errors of omission vs. errors of commission effect, What is the probability of winning the car by, What is the probability of winning the car, This page was last edited on 25 November 2020, at 11:53. Brouwer’s Fixed-Point Theorem via Hex*, 5.4. [3] In this case, there are 999,999 doors with goats behind them and one door with a prize. The Folk Theorem for average payoffs, Chapter 7. The host acts as noted in the specific version of the problem. Moreover, the host is certainly going to open a (different) door, so opening a door (which door unspecified) does not change this.

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