Bertrand's paradox (probability)

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	Bertrand's Paradox
		The Bertrand's Paradox is one such discovery that made mathematicians wary of the whole notion of probability.
		Find the probability that a chord chosen at random be longer than the side of an inscribed equilateral triangle.
		It appears that the midpoints of chords in a circle whose endpoints are uniformly distributed on the circumference are more densely located near its center than towards its periphery.
	www.cut-the-knot.com /bertrand.html

	Bertrand paradox (economics) - Wikipedia, the free encyclopedia
		There is another, different, Bertrand's paradox related to probability; see Bertrand's paradox (probability).
		The Bertrand paradox rarely appears in practice because real products are almost always differentiated in some way other than price (brand name, if nothing else), companies have limitations on their capacity to manufacture and distribute, and two companies rarely have identical costs.
		In economics, the Bertrand paradox describes a situation in which two players reaching a state of Nash equilibrium find themselves with no profits.
	en.wikipedia.org /wiki/Bertrand_paradox_(economics) (340 words)

	Math Forum - Ask Dr. Math
		Date: 05/09/99 at 23:00:45 From: David Krasik Subject: Bertrand's Paradox Dear Dr. Math, I've visited several math sites on Bertrand's Paradox and I've been to the library to look for information on it, but I haven't found anything that has helped me to clearly understand it.
		Date: 05/10/99 at 08:27:59 From: Doctor Anthony Subject: Re: Bertrand's Paradox Bertrand's Paradox shows that you can get 3 different answers to the question.
		My teacher showed it to my probability and statistics class basically to confuse us and my job is to unconfuse us.
	mathforum.org /library/drmath/view/56626.html (340 words)

	Bertrand's paradox (probability) - Wikipedia, the free encyclopedia
		Bertrand's paradox is a problem in probability theory.
		This article is about Bertrand's paradox in probability theory.
		There is another, different, Bertrand paradox related to economics; see Bertrand paradox (economics).
	en.wikipedia.org /wiki/Bertrand's_paradox_(probability) (600 words)

	Talk:Bertrand paradox (economics) - Wikipedia, the free encyclopedia
		I search for Bertrand's paradox on google and it appears that the paradox on probability is the more common find: that is what is the probability that a randomly picked chord of a circle will have a length greater than the side of an equilateral triangle inscribed in the circle.
		Hi, please consider the 2 entries in wikipedia (I am refering to the Bertrand's Paradox entry and the Bertrand Paradox entry).
		Or did Bertrand came up with 2 different paradoxes.
	www.wikipedia.org /wiki/Talk:Bertrand_paradox_%28economics%29 (384 words)

	Another Bertrand Paradox (ResearchIndex)
		Abstract: We report a surprising property of the classical Bertrand model with simultaneous entry: if the number of potential competitors is increased above two, the market breaks down with higher probability, and the competitive outcome becomes less likely.
		More potential competition lowers welfare --- another Bertrand paradox.
		1 Introduction One of the most inspiring paradoxes in industrial organization was Bertrand's contention --- sometimes paraphrased as "two is enough for competition" --- that...
	citeseer.nj.nec.com /elberfeld94another.html (384 words)

	One Hundred Interesting Mathematical Calculations and Puzzles, Number 10: Archive Entry From Brad DeLong's Webjournal
		While everyone is compiling puzzles involving probability and philosophy, one of the nicest puzzles in this area is Bertrand’s Paradox.
		The paradox is that you are almost certain to pick a lower number before a higher number, even though you choose randomly.
		I agree the paradox wouldn't be interesting if it was just another puzzle that arose by assuming that it is possible to pick an integer at random in a way that any integer is equally likely.
	www.j-bradford-delong.net /movable_type/archives/001395.html (2332 words)

	A Dynamic Model of Bertrand Competition with Entry
		"Bertrand Competition Under Uncertainty," CIRJE F-Series CIRJE-F-117, CIRJE, Faculty of Economics, University of Tokyo.
		We report a surprising property of the symmetric equilibrium solution: If the number of potential competitors is increased above two, the market breaks down with higher probability, and the competitive outcome becomes less likely.
		"A Comparison of Two-Market Bertrand Duopoly and Two-Market Cournot Duopoly," Working papers 2002-14, University of Connecticut, Department of Economics.
	www.ideas.uqam.ca /ideas/data/Papers/wpawuwpmi9701003.html (388 words)

	Another Bertrand Paradox (ResearchIndex)
		Abstract: We report a surprising property of the classical Bertrand model with simultaneous entry: if the number of potential competitors is increased above two, the market breaks down with higher probability, and the competitive outcome becomes less likely.
		1 Introduction One of the most inspiring paradoxes in industrial organization was Bertrand's contention --- sometimes paraphrased as "two is enough for competition" --- that...
		More potential competition lowers welfare --- another Bertrand paradox.
	citeseer.ist.psu.edu /elberfeld94another.html (287 words)

	Joseph Louis François Bertrand
		He is also famous for a paradox in the field of probability, now known as Bertrand's Paradox.
		Joseph Louis François Bertrand (March 11, 1822 - April 5, 1900, born and died in Paris) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, and thermodynamics.
		Bertrand was a professor at the École Polytechnique and the Collège de France.
	toshare.dynup.net /en/Joseph_Louis_Fran%c3%a7ois_Bertrand.htm (226 words)

	CTK Exchange
		Bertrand's Paradox ceased to be a paradox when it was realized that two-dimensional probability distributions need special handling.
		I am personally not convinced that there exists a single correct distribution that results in probability 1/3.
		I am well satisfied with the assertion that in two dimensions uniformity of a distribution is defined depending on the problem framework.
	www.cut-the-knot.com /exchange/bertrand2.shtml (67 words)

	5 April History: This Date
		His book Calcul des probabilités (1888) contains a paradox on continuous probabilities, Bertrand's paradox, which is that there are three solutions to the problem: Determine the probability that a random chord of a circle of unit radius has a length greater than the square root of 3 (the side of an inscribed equilateral triangle).
		Bertrand also worked on differential geometry and probability theory.
		In 1845 Bertrand conjectured that there is at least one prime between n and 2n-2 for every n > 3, which was proved by Chebyshev [16 May 1821 – 08 Dec 1894] in 1850.
	h42day.0catch.com /history/h4apr/h4apr05.html (11350 words)

	One Hundred Interesting Mathematical Calculations and Puzzles, Number 10: Archive Entry From Brad DeLong's Webjournal
		While everyone is compiling puzzles involving probability and philosophy, one of the nicest puzzles in this area is Bertrand’s Paradox.
		The paradox is that you are almost certain to pick a lower number before a higher number, even though you choose randomly.
		I agree the paradox wouldn't be interesting if it was just another puzzle that arose by assuming that it is possible to pick an integer at random in a way that any integer is equally likely.
	www.j-bradford-delong.net /movable_type/archives/001395.html (2332 words)

	Bertrand's paradox (probability) -- Facts, Info, and Encyclopedia article
		Bertrand's paradox is a problem in (The branch of applied mathematics that deals with probabilities) probability theory.
		According to the classical definition, the probability of a compound event is the ratio of the number of favorable cases to the total number of cases.
		The length of the arc is one third of the circumference of the circle, therefore the probability a random chord is longer than a side of the inscribed triangle is one third.
	www.absoluteastronomy.com /encyclopedia/B/Be/Bertrands_paradox_(probability).htm (641 words)

	Paradox, paradox examples, paradox for dos
		The Bertrand's Paradox is one such discovery that made mathematicians wary of the whole notion of probability.
		An overview of Shepard and Deutsch tones, the tritone paradox, and acoustic illusions analogous to spiral motion aftereffects is presented.
		This is not a true paradox since the poet may have knowledge that at least one...
	www.mainnewsletter.com /paradox.html (959 words)

	Paradox - Layman's Guide to the Banach-Tarski Paradox
		The Bertrand's Paradox is one such discovery that made mathematicians wary of the whole notion of probability.
		If you have any comments about Paradox or their songs, then please post them below: Paradox are awesome, very unique singing style but also very cool.
		A Paradox of Logic by Lewis Carroll, from the Platonic Realms Interactive Math Encyclopedia.
	webinfosites.com /q/paradox.htm (191 words)

	Bertrand's paradox (probability) - Wikipedia, the free encyclopedia
		Bertrand intended to show that the classical definition of probability is not applicable to a problem with an infinity of possible outcomes.
		According to the classical definition, the probability of a compound event is the ratio of the number of favorable cases to the total number of cases.
		The length of the arc is one third of the circumference of the circle, therefore the probability a random chord is longer than a side of the inscribed triangle is one third.
	en.wikipedia.org /wiki/Bertrand's_paradox_(probability) (191 words)

	berry-esseen_theorem
		Bayes' theorem Bayesian model comparison Bean machine Berkson's paradox Bernoulli trial Berry-EssÃ©en theorem Bertrand's paradox (probability) Binomial probability Boole's inequality Borel-Cantelli lemma C...
		Barbier's theorem, Bayes' theorem, Beatty's theorem, Beck's theorem, Bell's theorem, Berry-EssÃ©en theorem, Bertrand's ballot theorem, Binomial theorem, Bishop-Gromov inequality, Bolyai-Gerwien...
		the Team MATHWORLD - IN PRINT Order book from Amazon Probability and Statistics Â Â Moments Â Berry-EssÃ©en Theorem If is a probability distribution with zero mean and (1) where the above integral is a...
	berry-esseen_theorem.networklive.org (191 words)

	Maximum Entropy and the Glasses you are Looking Through
		We show that, using our procedure, one can properly solve at least three problems where Maximum Entropy fails: (1) Pearl's bell problem; (2) disjunctive constraints and (3), for continuous sample spaces, Bertrand's paradox.
		One of the most serious of these (known alternatively as Pearl's, Dalkey's or Hunter's `bell problem') can occur if Maximum Entropy is applied to Bayesian networks.
		It can be applied in all domains involving probability distributions that are only partially specified.
	www.cs.berkeley.edu /Seminar.archive/1999/09.Sep/990914.grunwald.html (163 words)

	Paradox - Wikipedia, the free encyclopedia
		Bertrand's paradox (probability): different common-sense defintions of randomness give quite different results
		Thus, the paradox of Frederic's birthday in The Pirates of Penzance establishes the surprising fact that a person may be more than N years old on his Nth birthday.
		Arrow's paradox/Voting paradox/Condorcet paradox: You can't have all the attributes of an ideal voting system at once
	phatnav.com /wiki/index.php?title=Paradox (163 words)

	Paradox - Wikipedia, the free encyclopedia
		Bertrand's paradox (probability): Different common-sense definitions of randomness give quite different results.
		Unexpected hanging paradox: The day of the hanging will be a surprise, so it cannot happen at all, so it will be a surprise.
		Zeno's paradoxes: "You will never reach point B from point A as you must always get half-way there, and half of the half, and half of that half, and so on..."
	en.wikipedia.org /wiki/Paradox (2330 words)

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