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Boy Or Girl Paradox
The Boy or Girl paradox surrounds a set of questions in probability theory, which are also known as The Two Child Problem, Mr. Smith's Children and the Mrs. Smith Problem. The initial formulation of the question dates back to at least 1959, when Martin Gardner featured it in his October 1959 " Mathematical Games column" in ''Scientific American''. He titled it The Two Children Problem, and phrased the paradox as follows: *Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls? *Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys? Gardner initially gave the answers and , respectively, but later acknowledged that the second question was ambiguous. Its answer could be , depending on the procedure by which the information "at least one of them is a boy" was obtained. The ambiguity, depending on the exact wording and possible assumptions, was confirmed by Maya Bar-Hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Process
In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables ''X''''i'' are identically distributed and independent. Prosaically, a Bernoulli process is a repeated coin flipping, possibly with an unfair coin (but with consistent unfairness). Every variable ''X''''i'' in the sequence is associated with a Bernoulli trial or experiment. They all have the same Bernoulli distribution. Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes (such as the process for a six-sided die); this generalization is known as the Bernoulli scheme. The problem of determining the process, given only a limited sample of Bernoulli trials, may be called the problem of checking whether a coin is fair. Definition A Bernoulli process is a f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Paradoxes
This list includes well known paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. This list collects only scenarios that have been called a paradox by at least one source and have their own article in this encyclopedia. Although considered paradoxes, some of these are simply based on fallacious reasoning ( falsidical), or an unintuitive solution ( veridical). Informally, the term ''paradox'' is often used to describe a counter-intuitive result. However, some of these paradoxes qualify to fit into the mainstream perception of a paradox, which is a self-contradictory result gained even while properly applying accepted ways of reasoning. These paradoxes, often called ''antinomy,'' point out genuine problems in our understanding of the ideas of truth and description. Logic * : The supposition that, 'if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved' leads to paradoxica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two Envelopes Problem
The two envelopes problem, also known as the exchange paradox, is a paradox in probability theory. It is of special interest in decision theory, and for the Bayesian interpretation of probability theory. It is a variant of an older problem known as the necktie paradox. The problem is typically introduced by formulating a hypothetical challenge like the following example: Since the situation is symmetric, it seems obvious that there is no point in switching envelopes. On the other hand, a simple calculation using expected values suggests the opposite conclusion, that it is always beneficial to swap envelopes, since the person stands to gain twice as much money if they switch, while the only risk is halving what they currently have. Introduction Problem Basic setup: A person is given two indistinguishable envelopes, each of which contains a sum of money. One envelope contains twice as much as the other. The person may pick one envelope and keep whatever amount it contains. They ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Necktie Paradox
The necktie paradox is a puzzle and paradox with a subjective interpretation of probability theory describing a paradoxical bet advantageous to both involved parties. The two-envelope paradox is a variation of the necktie paradox. Statement of paradox Two persons, each given a necktie, start arguing over who has the cheaper one. The person with the more expensive necktie must give it to the other person. The first person reasons as follows: winning and losing are equally likely. If I lose, then I will lose the value of my necktie. But if I win, then I will win more than the value of my necktie. Therefore, the wager is to my advantage. The second person can consider the wager in exactly the same way; thus, paradoxically, it seems both persons have the advantage in the bet. Resolution using fluid intelligence The paradox can be resolved by giving more careful consideration to what is lost in one scenario ("the value of my necktie") and what is won in the other ("more than the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertrand Paradox (probability)
The Bertrand paradox is a problem within the classical interpretation of probability theory. Joseph Bertrand introduced it in his work ''Calcul des probabilités'' (1889), as an example to show that the principle of indifference may not produce definite, well-defined results for probabilities if it is applied uncritically when the domain of possibilities is infinite. Bertrand's formulation of the problem The Bertrand paradox is generally presented as follows: Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle? Bertrand gave three arguments (each using the principle of indifference), all apparently valid, yet yielding different results: # The "random endpoints" method: Choose two random points on the circumference of the circle and draw the chord joining them. To calculate the probability in question imagine the triangle rotated so its vertex coi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heuristics
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, short-term goal or approximation. Where finding an optimal solution is impossible or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution. Heuristics can be mental shortcuts that ease the cognitive load of making a decision. Examples that employ heuristics include using trial and error, a rule of thumb or an educated guess. Heuristics are the strategies derived from previous experiences with similar problems. These strategies depend on using readily accessible, though loosely applicable, information to control problem solving in human beings, machines and abstract issues. When an individual applies a heuristic in practice, it generally performs as expected. However it can alternatively c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertrand's Box Paradox
Bertrand's box paradox is a veridical paradox in elementary probability theory. It was first posed by Joseph Bertrand in his 1889 work Calcul des Probabilités'. There are three boxes: # a box containing two gold coins, # a box containing two silver coins, # a box containing one gold coin and one silver coin. The question is to calculate the probability, after choosing a box at random and withdrawing one coin at random, if that happens to be a gold coin, of the next coin drawn from the same box also being a gold coin. A veridical paradox is when the correct solution to a puzzle appears to be counterintuitive. It may seem intuitive that the probability that the remaining coin is gold should be , but the probability is actually . However, this is not the paradox Bertrand referred to. He showed that if were correct, it would lead to a contradiction, so cannot be correct. This simple but counterintuitive puzzle is used as a standard example in teaching probability theory. The s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monty Hall Problem
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. The problem was originally posed (and solved) in a letter by Steve Selvin to the ''American Statistician'' in 1975. It became famous as a question from reader Craig F. Whitaker's letter quoted in Marilyn vos Savant's "Ask Marilyn" column in '' Parade'' magazine in 1990: Vos Savant's response was that the contestant should switch to the other door. Under the standard assumptions, the switching strategy has a probability of winning the car, while the strategy that remains with the initial choice has only a probability. When the player first makes their choice, there is a chance that the car is behind one of the doors not chosen. This probability does not change after the host reveals a goat behind one of the unchosen doors. When the host provides information about the 2 unchosen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Experimental Psychology
The ''Journal of Experimental Psychology'' was a bimonthly peer-reviewed academic journal published by American Psychological Association. The journal, established in 1916, contained articles relating to experimental psychology. Beginning in 1975, three independently edited and distributed sections were split off with an additional section being added in 1995. The ''Journal of Experimental Psychology'' was renamed '' Journal of Experimental Psychology: General''. History The first issue of the journal was published by the Psychological Review Company, Princeton, New Jersey Princeton is a municipality with a borough form of government in Mercer County, in the U.S. state of New Jersey. It was established on January 1, 2013, through the consolidation of the Borough of Princeton and Princeton Township, both of w .... The following journals are currently published as independently edited and distributed sections of the former ''Journal of Experimental Psychology'': * '' Journ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marilyn Vos Savant
Marilyn vos Savant (; born Marilyn Mach; August 11, 1946) is an American magazine columnist who has the highest recorded intelligence quotient (IQ) in the ''Guinness Book of Records'', a competitive category the publication has since retired. Since 1986, she has written "Ask Marilyn", a ''Parade'' magazine Sunday column wherein she solves puzzles and answers questions on various subjects, and which popularized the Monty Hall problem in 1990. Biography Marilyn vos Savant was born Marilyn Mach on August 11, 1946, in St. Louis, Missouri, to parents Joseph Mach and Marina vos Savant. Savant says one should keep premarital surnames, with sons taking their fathers' and daughters their mothers'. The word ''savant'', meaning someone of learning, appears twice in her family: her grandmother's name was Savant; her grandfather's, vos Savant. She is of Italian, Czechoslovak, German, and Austrian ancestry, being descended from the physicist and philosopher Ernst Mach. As a teenager, Savan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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