Bertrand's Problem
What is the Probability that a Chord drawn at Random on a Circle ofRadius 
has length 
? The answer, it turns out, depends on the interpretation of ``two points drawn atRandom.'' In the usual interpretation that Angles 
and 
are picked at Random on the Circumference,
However, if a point is instead placed at Random on a Radius of the Circle and aChord drawn Perpendicular to it,
The latter interpretation is more satisfactory in the sense that the result remains the same for a rotated Circle, aslightly smaller Circle Inscribed in the first, or for a Circle of the same size but with its centerslightly offset. Jaynes (1983) shows that the interpretation of ``Random'' as a continuousUniform Distribution over the Radius is the only one possessing all these three invariances.
References
Bogomolny, A. ``Bertrand's Paradox.'' http://www.cut-the-knot.com/bertrand.html. Jaynes, E. T. Papers on Probability, Statistics, and Statistical Physics. Dordrecht, Netherlands: Reidel, 1983. Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 42-45, 1995.
- Ramanujan's Sum
- Ramanujan's Sum Identity
- Ramanujan's Tau-Dirichlet Series
- Ramanujan's Tau Function
- Ramanujan Theta Functions
- Ramp Function
- Ramphoid Cusp
- Ramsey Number
- Ramsey's Theorem
- Randelbrot Set
- Random Distribution
- Random Dot Stereogram
- Random Graph
- Random Matrix
- Random Number
- Random Percolation
- Random Polynomial
- Random Variable
- Random Walk
- Random Walk--1-D
- Random Walk--2-D
- Random Walk--3-D
- Range (Image)
- Range (Line Segment)
- Range (Statistics)