I recently watched Numberphile's excellent videos about Bertrand's paradox.
See also https://en.wikipedia.org/wiki/Bertrand_paradox_(probability) for a description.
At first, please excuse my ignorance. This is my second post about this subject, the first one was flawed and has since been removed. I don’t speak the math language but I do have a sincere wish to understand what I can learn from this “paradox”. Should it be that this post is flawed too, please explain why (as pedagogically as possible), close it and there will not be a third :).
To me, the “choose a random xxx” obscures what I believe to be the core of Bertrand’s paradox, namely different spaces and functions to generate new spaces from existing ones. My understanding is that the act of “choosing a random xxx” first requires a well defined space to “choose an xxx” from, and that we hence can disregard the randomising aspects and instead focus on analysing the underlying spaces and functions.
Unless it is deemed to be a completely pointless exercise, could someone please help me identify and express the involved spaces and functions in a rigorous form? My hope is that this would help me to better understand what I believe is the core of the “paradox”. Below are my own naive attempts.
Definitions (For a given circle and simplified by assuming a fixed rotation.)
Space A: The space of all arc lengths*. All real numbers from 0 to the circumference/2. PDF assumed (or defined?) to be uniform. This is the “starting” space for Space C below.
Space B: The space of all distances from the origin along a radius. All real numbers from 0 to the radius. PDF assumed (or defined?) to be uniform. This is the “starting” space for Space D below.
Space C: The space of all chords that can be created from Space A by applying the function “chord length as a function of arc length” to each element in Space A. The PDF is well defined. One property of the PDF is that P(chord length > sqrt(3))=1/3.
Space D: The space of all chords that can be created from Space B by applying the function “chord length as a function of distance from the origin” to each element in Space B. The PDF is well defined. One property of the PDF is that P(chord length > sqrt(3))=1/2.
Can Space C and Space D be considered to be “equals”/"peers" under the axioms and rules of set/space theory or are we comparing apples and oranges?
Is there a reason to believe that a "Space E" that contains "all chords of a circle" can be created from another Space?
Are we "allowed" to define a "Space E" to contain "all chords of a circle"?
*Assuming that “choose two random points on the circle” is equivalent to “choose a random arc length”.