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Bertrand's paradox extended. A chord of the unit circle is picked at random. What is the probability that an equilateral triangle with the chord as base can fit inside the circle if: (a) the chord passes through a point P picked uniformly in the disk, and the angle it makes with a fixed direction is uniformly distributed on [0, 27), (b) the chord passes through a point P picked uniformly at random on a randomly chosen radius, and the angle it makes with the radius is uniformly distributed on [0, 27).

Bertrand's paradox extended. A chord of the unit circle is picked at random. What is the probability that an equilateral triangle with the chord as base can fit inside the circle if: (a) the chord passes through a point P picked uniformly in the disk, and the angle it makes with a fixed direction is uniformly distributed on [0, 27), (b) the chord passes through a point P picked uniformly at random on a randomly chosen radius, and the angle it makes with the radius is uniformly distributed on [0, 27).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 74E
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Bertrand's paradox extended. A chord of the unit circle is picked at random. What is the probability that an equilateral triangle with the chord as base can fit inside the circle if: (a) the chord passes through a point P picked uniformly in the disk, and the angle it makes with a fixed direction is uniformly distributed on [0, 27), (b) the chord passes through a point P picked uniformly at random on a randomly chosen radius, and the angle it makes with the radius is uniformly distributed on [0, 27).
Transcribed Image Text:Bertrand's paradox extended. A chord of the unit circle is picked at random. What is the probability that an equilateral triangle with the chord as base can fit inside the circle if: (a) the chord passes through a point P picked uniformly in the disk, and the angle it makes with a fixed direction is uniformly distributed on [0, 27), (b) the chord passes through a point P picked uniformly at random on a randomly chosen radius, and the angle it makes with the radius is uniformly distributed on [0, 27).
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