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Suppose that somebody asks us the following question:

Consider a straight line segment that goes from 0 to 1 (inclusive). Suppose that a point is chosen at random on this line segment. What is the probability that the point corresponds to a number greater than 0.5?

The intuitive answer to this question is obviously 0.5, but somebody comes along and disagrees. He claims that it is possible to randomly choose a point along the line segment in such a way that the answer is actually 0.76. This is how he does it:

First he chooses a number $x$ from 0 to 9 (inclusive) subject to a uniform probability density function (PDF) of $f(x)=1/9$. Next he takes $x$ and transforms it according to the equation $y=\log_{10}(x+1)$. The PDF of $y$ is $g(y)=\ln(10)*10^y/9$, and $\int_{0.5}^1g(y)dy\approx0.76$. When you tell him that this conclusion is patently absurd he replies that there are an infinite number of ways of randomly choosing a point on the line segment so your answer to the original question is no better than his.

So who is right? Clearly if the word "random" in the original question refers to any PDF that covers the line segment then he is right. But our intuition tells us that the word "random" here actually refers to a uniform PDF that covers the line segment. In other words, the points on the line should be uniformly distributed, with no subdivision of the line segment of a certain length containing more points on average than any other subdivision of the same length. This uniform distribution of points on the line segment is then subject to some external condition, which some fraction of the points satisfies and the remainder does not. A non-uniform PDF that covers the line segment and causes a greater proportion (and sometimes a much greater proportion) of the points to satisfy the external condition seems horribly contrived and contrary to our intuitive notion of "random" meaning "uniform"; a distribution that favors no location over another.

Similarly, if instead of a "random" distribution of points over a line segment we were instead talking about a "random" distribution of objects over an area we would expect that distribution to be uniform; no sub-area would contain more objects on average than any other sub-area of the same size. And if we were talking about a "random" distribution of objects over a volume then we would also expect that distribution to be uniform; no sub-volume would contain more objects on average than any other sub-volume of the same size.

In the light of this analysis let us look at Bertrand's Paradox:

Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?

Classically there are three different ways given of "randomly" choosing the chords: (1) the "random endpoint" method (yielding an answer of 0.33), (2) the "random radius" method (yielding an answer of 0.5), and (3) the "random midpoint" method (yielding an answer of 0.25). Although all of these methods fully cover the circle with chords, only the "random radius" method produces a uniform distribution of chords in which no sub-area of the circle contains more chords on average than any other sub-area of the circle of the same size. The other two methods place more chords in certain sub-areas of the circle than in other sub-areas of the same size, which changes the proportion of chords which satisfy the external condition of being longer than a side of the triangle. More specifically, both methods (1) and (3) place more chords closer to the circumference of the circle than method (2), resulting in a reduction in the proportion of chords longer than a side of the triangle. A visualization of the chord distribution produced by methods (1)-(3) can be seen below (from Wikipedia):

Chord Distributions

Notice that even though methods (1) and (3) both produce non-uniform PDFs of chords over the interior of the circle, they are both created as mappings from uniform PDFs of some kind. For example, in the case of method (1) a uniform PDF of two points over the circumference of the circle maps onto a non-uniform PDF of chords over the interior of the circle. In the case of method (3) a uniform PDF of midpoints over the interior of the circle maps onto a non-uniform PDF of chords over the interior of the circle.

Both methods (1) and (3) are akin to the guy starting from a uniform PDF of $f(x)=1/9$ and mapping it onto a non-uniform PDF of points over the line segment $g(y)=\ln(10)*10^y/9$, instead of choosing a uniform PDF of points over the line segment to begin with. If we are to be consistent with our intuitions in both of these cases then we should choose method (2) by default, since only this method yields a uniform PDF of the objects we are actually interested in over the space we are actually interested in -- namely, chords over the interior of the circle. More specifically, if we wish to preserve the basic intuition that the word "random" in these types of questions implies a uniform PDF then method (2) must be the default method of selecting a chord -- unless a non-uniform PDF is explicitly required instead. This resolves Bertrand's Paradox, and the answer to Bertrand's riddle is 0.5.

Is there any flaw in this reasoning?

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  • $\begingroup$ Is your question solely about the Bertrand paradox case, and you merely gave the rest of the text above for context? Anyway, to me the point is simpler: whenever you say "uniform" you must clarify "uniform over what?" That is, Bertrand's paradox isn't answered by "this is the best method" but by "your question isn't complete because you haven't said what "uniform" means". $\endgroup$ – Ian Mar 5 '17 at 17:58
  • $\begingroup$ The point I'm trying to make is that the first question is very unambiguous while the second question (Bertrand's Paradox) is very ambiguous to many people, despite the fact that in both cases the exact PDF to be used is never specified. I propose a resolution to this ambiguity whereby "random" always implies a uniform PDF (over whatever space is specified by the question). $\endgroup$ – The Riddler Mar 5 '17 at 18:10
  • $\begingroup$ Sure, but then the question is "what is the natural choice of the space?" In the first case the space is just a line segment, but in the second case you could parametrize matters in several different ways. A similar issue occurs with the uniform distribution on a sphere: the surface area measure is not the same as the uniform distribution in the longitude and latitude. $\endgroup$ – Ian Mar 5 '17 at 18:12
  • $\begingroup$ WISDOM: (1) Intuition is a two bladed knife. (2) Being a Linguist in math is a bitter butter (and bread). Seriously: Math is working with models and many pseudo paradoxes like the Bertrand paradox and, say, the birthday paradox exemplify that our evolution driven intuition fails many times when it comes to probability. Geometry is another nice example, we wouldn't have ever discovered non-Euclidean geometry if we had been stuck to our commonsense intuition. $\endgroup$ – zoli Mar 5 '17 at 18:19
  • $\begingroup$ As I demonstrated though you can parametrize a line segment in an infinite number of ways as well, but only one of these parametrizations (the uniform one) corresponds to our intuition of a "random" distribution. In Bertrand's Paradox we get confused because (for instance, in the case of method (1)) a uniform PDF over the circumference of the circle produces a non-uniform PDF over the interior of the circle, and the latter distribution is the one that is actually relevant here. $\endgroup$ – The Riddler Mar 5 '17 at 18:22

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