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MENTAL MODEL · M5719

Bertrand Paradox

Bertrand Paradox
TechnicalHigh supportProbability Theory
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Version 1.0.0 · Updated 2026-07-30

CORE DEFINITION

“What is the probability that a randomly drawn chord in a circle is longer than the side of an inscribed equilateral triangle?” Depending on the definition of “random” (random endpoints, random radius, random center), three correct answers can be obtained: 1/3, 1/2, and 1/4.

SCAFFOLDING EFFECT

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The ambiguity of “random.” When we say “random sampling,” if the specific generative mechanism (geometric process) of sampling is not clearly specified, “probability” is undefined. Many debates in data analysis stem from different understandings of “uniform distribution.”

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The same “randomly draw a chord” problem yields three answers (1/3, 1/2, 1/4) due to three generative mechanisms of “random” (random endpoints/radius/center); probability depends on the sampling definition.

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Source support: Explicit

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    en.wikipedia.orghttps://en.wikipedia.org/wiki/Bertrand%27s_paradoxZH · Explicit
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