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):
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?