Bertrand's paradox

Probabilities presuppose categories. The probability of throwing a "4" with a normal dice is 1/6 (or 16.6%) because a normal dice has six faces that each represent another number category. For a dice with two "2"s, two "4"s, and two "6"s, however, the  probability of throwing a "4" is 1/3 (or 33.3%) because such a dice represents only three number categories.

Usually, people have no problem with the probabilities involved in simple dice throwing. In general, however, people have a poor intuition for everyday probabilities when things become slightly more complicated --- Test your intuition

Often, people are hardly aware that they use categories to estimate probabilities. Consider, for instance, the following four outcomes of throwing two sticks randomly on the floor:


Most people would say intuitively that the outcome at the left has a high probability and the outcome at the right a low probability. This, however, is true only if one takes each of these four outcomes as a representative of a category of similar outcomes. Without presupposing such categories, every outcome would have the same probability!

But, then, what are the categories? Why would the four outcomes belong to different categories? Could yet further categories be distinguished? How many individual outcomes fall in a distinguishable category? Only when these questions have been given an answer, a probability can be assigned to an individual outcome.

These questions, however, may not have clear-cut answers at all. In the 19th century, Joseph Louis Bertrand (1889) realized that this is a fundamental problem to probability theory. He formulated this problem in the form of a paradox that is illustrated by the following riddle:

Suppose the outer circle has a radius of 1, and the inner disk a radius of 0.5. Question: What is the probability that a randomly picked outer-circle chord crosses the inner disk? (A chord, like here the ones in red, is a straight line between two points on a circle.)

This riddle seems straightforward but can only be answered by a parameterization (a structural description) of the chords, which implies a categorization of the chords into different categories. Then, however, different categorizations appear to yield different answers!

Answer 1		Answer 2
Assume a uniform distribution over the midpoint positions of parallel chords.		Assume a uniform distribution over the orientations of chords with the same starting point.
Half of these chords cross the inner disk, so, picking such a chord has a probability of 0.50.		One-third of these chords cross the inner disk, so, picking such a chord has a probability of 0.33.

If one excludes chords crossing the circle's center, there is even a third answer (a probability of 0.25). One may have compelling arguments to choose a specific categorization, but the point is that no categorization can be proved to be the one and only "true" one.

Hence, probabilities depend on the chosen categorization of the things involved, which is obtained by way of structural descriptions of those individual things. In other words, structural descriptions precede probabilities: one first needs a description language that produces categories and, only then, probabilities can be determined.

In perceptual organization, this means that one needs a structural description language that produces the same categories of stimulus interpretations as those produced by the human visual system. Only then, one may assess the probabilities with which specific interpretations are predicted to be perceived. Notice that, by Bertrand's paradox, one can never prove that these perceptual probabilities agree with "true" probabilities of occurrence of things in the world.


For related demos, see T-junctions,  Object versus viewer, and Occam, von Helmholtz, and Bayes

For an extensive discussion on these issues, see Psychological Bulletin 2000
For an updated brief discussion on these issues, see Acta Psychologica 2011