[58][59] Three cards from an ordinary deck are used to represent the three doors; one 'special' card represents the door with the car and two other cards represent the goat doors. Monty is saying in effect: you can keep your one door or you can have the other two doors, one of which (a non-prize door) I'll open for you." Raquel has to choose whether to pursue training that costs $1;000 to herself or not. flourishing subject of game theory. The question is whether knowing the warden's answer changes the prisoner's chances of being pardoned. [4] Due to the overwhelming response, Parade published an unprecedented four columns on the problem. Length of play for random-turn Recursive Majority, Chapter 10. 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. 1/3 must be the average probability that the car is behind door 1 given the host picked door 2 and given the host picked door 3 because these are the only two possibilities. The 2/3 chance of finding the car has not been changed by the opening of one of these doors because Monty, knowing the location of the car, is certain to reveal a goat. The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show Let's Make a Deal and named after its original host, Monty Hall. As in the Monty Hall problem, the intuitive answer is 1/2, but the probability is actually 2/3. Vos Savant asks for a decision, not a chance. [5] Paul Erdős, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation demonstrating vos Savant's predicted result.[6]. After a reader wrote in to correct the mathematics of Adams's analysis, Adams agreed that mathematically he had been wrong. 2?" I personally read nearly three thousand letters (out of the many additional thousands that arrived) and found nearly every one insisting simply that because two options remained (or an equivalent error), the chances were even. reader to the blend of economic insight, mathematical elegance, yet entertaining manner. − [1][2] It became famous as a question from a reader's letter quoted in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990:[3]. Your choice of door A has a chance of 1 in 3 of being the winner. Is it to your advantage to switch your choice?[9]. [14][15][16][17][18] As Cecil Adams puts it,[14] "Monty is saying in effect: you can keep your one door or you can have the other two doors." Nalebuff, as later writers in mathematical economics, sees the problem as a simple and amusing exercise in game theory. "Game -- Gábor Lugosi, Pompeu Fabra University, [23], Most statements of the problem, notably the one in Parade Magazine, do not match the rules of the actual game show [11] and do not fully specify the host's behavior or that the car's location is randomly selected. alike, this book is a must. In the article, Hall pointed out that because he had control over the way the game progressed, playing on the psychology of the contestant, the theoretical solution did not apply to the show's actual gameplay. He then says to you, "Do you want to pick door No. 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. And the chance aspects of how the car is hidden and how an unchosen door is opened are unknown. Strategic dominance links the Monty Hall problem to the game theory. 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. [19] Numerous examples of letters from readers of Vos Savant's columns are presented and discussed in The Monty Hall Dilemma: A Cognitive Illusion Par Excellence. After the player picks his card, it is already determined whether switching will win the round for the player. [3] Though vos Savant gave the correct answer that switching would win two-thirds of the time, she estimates the magazine received 10,000 letters including close to 1,000 signed by PhDs, many on letterheads of mathematics and science departments, declaring that her solution was wrong. In general, the answer to this sort of question depends on the specific assumptions made about the host's behavior, and might range from "ignore the host completely" to "toss a coin and switch if it comes up heads"; see the last row of the table below. elegance. Pure optimal strategies: Saddle points, 2.5. Whether you change your selection or not, the odds are the same. So, in this particular expression, the choosing of the host doesn't depend on where the car is, and there's only two remaining doors once X1 is chosen (for instance, P(H1|X1) = 0); and P(Ci,Xi) = P(Ci)P(Xi) because Ci and Xi are independent events (the player doesn't know where is the car in order to make a choice). theory as a field.

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