Comments by "EebstertheGreat" (@EebstertheGreat) on "More on Bertrand's Paradox (with 3blue1brown) - Numberphile" video.

  1. It's worth pointing out to people who don't know that the mathematics of probability is not in dispute at all. It is not possible to phrase an ambiguous question like this using formal mathematical symbols only. The paradox comes from the interpretation of the math. In particular, there is no mathematical meaning to picking something "at random." Rather, we can describe random variables, which are measurable functions from probability spaces to measurable spaces. If we try to encode the question in Bertrand's paradox formally, most of it can be done unambiguously, but there isn't enough information to determine the distribution of the random variable. Specifically, we want a random variable X which takes elements of the probability triple (Ω, F, P) to a measurable space (E, 𝓔). Here, E is the set of chords of the circle and 𝓔 is a σ-algebra on E. Ω represents the set of possible parameters or "inputs" to decide how to throw, with F a σ-algebra on Ω and P a probability measure on (Ω,F). To say that X is "uniform on (E, 𝓔)," we mean that for all U, V ∈ 𝓔, P(X ∈ U) = P(X ∈ V) iff U and V are in some sense the "same size." To determine that, we need a measure on (E, 𝓔), and that's where we run into problems here. People disagree about what the most "natural" measure is on the set of chords of the unit circle. (There is also disagreement on how to define the "inputs," but it doesn't actually matter here, since the definition of "uniform" doesn't depend on that.) So mathematically speaking, the problem really is just ill-posed. Philosophically speaking, maybe some people think we should expect to have some better way to handle this mathematically, but currently at least, we just don't.
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