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تسجيل مستخدم جديدPrize insights in probability, and one goat of a recycled error: Jason Rosenhouses The Monty Hall Problem
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Anthony B. Morton
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The Monty Hall problem is the TV game scenario where you, the contestant, are presented with three doors, with a car hidden behind one and goats hidden behind the other two. After you select a door, the host (Monty Hall) opens a second door to reveal a goat. You are then invited to stay with your original choice of door, or to switch to the remaining unopened door, and claim whatever you find behind it. Assuming your objective is to win the car, is your best strategy to stay or switch, or does it not matter? Jason Rosenhouse has provided the definitive analysis of this game, along with several intriguing variations, and discusses some of its psychological and philosophical implications. This extended review examines several themes from the book in some detail from a Bayesian perspective, and points out one apparently inadvertent error.
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The Monty Hal problem is an attractive puzzle. It combines simple statement with answers that seem surprising to most audiences. The problem was thoroughly solved over two decades ago. Yet, more recent discussions indicate that the solution is incomp
letely understood. Here, we review the solution and discuss pitfalls and other aspects that make the problem interesting.
The rational solution of the Monty Hall problem unsettles many people. Most people, including the authors, think it feels wrong to switch the initial choice of one of the three doors, despite having fully accepted the mathematical proof for its super
iority. Many people, if given the choice to switch, think the chances are fifty-fifty between their options, but still strongly prefer to stay with their initial choice. Is there some sense behind these irrational feelings? We entertain the possibility that intuition solves the problem of how to behave in a real game show, not in the abstract textbook version of the Monty Hall problem. A real showmaster sometimes plays evil, either to make the show more interesting, to save money, or because he is in a bad mood. A moody showmaster erases any information advantage the guest could extract by him opening other doors which drives the chance of the car being behind the chosen door towards fifty percent. Furthermore, the showmaster could try to read or manipulate the guests strategy to the guests disadvantage. Given this, the preference to stay with the initial choice turns out to be a very rational defense strategy of the shows guest against the threat of being manipulated by its host. Thus, the intuitive feelings most people have about the Monty Hall problem coincide with what would be a rational strategy for a real-world game show. Although these investigations are mainly intended to be an entertaining mathematical commentary on an information-theoretic puzzle, they touch on interesting psychological questions.
The Coupon Collectors Problem is one of the few mathematical problems that make news headlines regularly. The reasons for this are on one hand the immense popularity of soccer albums (called Paninimania) and on the other hand that no solution is know
n that is able to take into account all effects such as replacement (limited purchasing of missing stickers) or swapping. In previous papers we have proven that the classical assumptions are not fulfilled in practice. Therefore we define new assumptions that match reality. Based on these assumptions we are able to derive formulae for the mean number of stickers needed (and the associated standard deviation) that are able to take into account all effects that occur in practical collecting. Thus collectors can estimate the average cost of completion of an album and its standard deviation just based on elementary calculations. From a practical point of view we consider the Coupon Collectors problem as solved. ----- Das Sammelbilderproblem ist eines der wenigen mathematischen Probleme, die regelma{ss}ig in den Schlagzeilen der Nachrichten vorkommen. Dies liegt einerseits an der gro{ss}en Popularitat von Fu{ss}ball-Sammelbildern (Paninimania genannt) und andererseits daran, dass es bisher keine Losung gibt, die alle relevanten Effekte wie Nachkaufen oder Tauschen berucksichtigt. Wir haben bereits nachgewiesen, dass die klassischen Annahmen nicht der Realitat entsprechen. Deshalb stellen wir neue Annahmen auf, die die Praxis besser abbilden. Darauf aufbauend konnen wir Formeln fur die mittlere Anzahl benotigter Bilder (sowie deren Standardabweichung) ableiten, die alle in der Praxis relevanten Effekte berucksichtigen. Damit konnen Sammler die mittleren Kosten eines Albums sowie deren Standardabweichung nur mit Hilfe von elementaren Rechnungen bestimmen. Fur praktische Zwecke ist das Sammelbilderproblem damit gelost.
The Secret Santa ritual, where in a group of people every member presents a gift to a randomly assigned partner, poses a combinatorial problem when considering the probabilities involved in the formation of pairs, where two persons exchange gifts mut
ually. We give different possible derivations for such probabilities by counting fixed-point-free permutations with certain numbers of 2-cycles.
Are you having trouble getting married? These days, there are lots of products on the market for dating, from apps to websites and matchmakers, but we know a simpler way! Thats right -- your path to coupled life isnt through Tinder: its through Sudok
u! Read our fabulous paper where we explore the Stable Marriage Problem to help you find happiness and stability in marriage through math. As a bonus, you get two Sudoku puzzles with a new flavor.
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