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Bertrand Paradox

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The Bertrand Paradox is a prime example of how two words—”random” and “equal probability” are incorrectly used to mean the same thing. The paradox shows that if you don’t define your probabilities well, then the mechanism that generates random variables will also not be well defined (Chang, 2012), leading to a paradox.

The development of Bertrand’s paradox was one of the reasons classical probability fell out of favor at the turn of the 20th century (Shafer and Vovk, 2006). However, it really isn’t a paradox at all: if you’re careful to define your probability spaces, then the paradox doesn’t exist, as the example below illustrates.

Bertrand Paradox Origins

In his 1889 treatise Calculs des Probabilities (pp. 6-7, as cited in Gorroochurn, 2012), Bertrand asked:

“If we choose at random two points on the surface of a sphere, what is the probability that the distance between them is less than 10?”

Bertrand offered two solutions, the first of which was that the probability a certain segment of the sphere’s surface, containing the two points, is proportional to the segment’s area as a whole. The second solution: a given arc on a great circle , with the two points, is proportional to the length of the arc. In fact, there are several ways to approach the problem, leading to multiple solutions and multiple probabilities.

Bertrand Paradox Example

To illustrate the paradox from a slightly different angle (pun intended), let’s take a look at an equilateral triangle drawn within a circle.
bertrand paradox

What we want to know is, if we pick a random chord (a line segment drawn between two points on the circumference), what are the odds the chord is longer than one side of the triangle?

The solution seems simple enough. Pick a random point in the circle, use that as a midpoint for a chord:
example paradox

This way of choosing a random chord gives a probability of 0.25 (Bartlett, 2014). But this obviously isn’t the only way to find a chord. We could choose a point on the perimeter to start with (that will give a probability of 2/3) or pick a random radius, then a random point (probability = ½).


All solutions are valid. However, they can’t all be valid at the same time—hence the paradox. The reason for the paradox is that the initial problem is ill-posed. We weren’t given any rules for choosing our chords, so that left the door open for multiple interpretations.

References

Bartlett, P. (2014). Lecture 2: Bertrand’s Paradox. Retrieved June 2, 2020 from: http://web.math.ucsb.edu/~padraic/ucsb_2013_14/math7h_s2014/math7h_s2014_lecture2.pdf
Gorroochurn, P. Classic Problems of Probability. 2012.
Chang, M. Paradoxes in Scientific Inference. CRC Press, 2012.
Sainsbury, R. Paradoxes. Cambridge University Press. 2009.
Shafer and Vovk, 2006

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Stephanie Glen. "Bertrand Paradox" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/bertrand-paradox/
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