Math (probability) puzzle

[Max™]

That's a bit further past Banach-Tarski regarding the axiom of choice than I've spent much time, and way deeper than the bit of probability theory that I know, so I'll take your word for it there atm. I'm more into geometry than probability. As I said, the main thing that was making my brain glitch was seeing the indifference principle brought up in what I couldn't discern as a Bayesian setup.

Something in the back of my head kept saying "but you can sample from all the bins, unless he said you can't, did he say you can't, wtf is going on here" lol.


Hell, that's why I clicked on it, actually, been poking at spots I don't feel as comfortable with in my math knowledge, and learned a bit more about probabilities than I knew before, yay.

[joft]

What I said was not deep, it's just the fact that a countable set has Lebesgue measure zero (it is the countable union of singletons and each singleton has measure zero), and a countable union of countable sets is countable (hence a countable union of a countable union of countable sets is also countable)

The point is just that you can't be a strict empiricist or frequentist or whatever and talk about probability on real number line.

[Max™]

Indeed, I hadn't considered that about the probability on the real line, which is what I learned. :D

[Necronus]

Wow, this thread scares me. Not in the way a thread about a serial killer might, but in a way that makes me wonder how much some people overthink a problem. What did your math teachers do to you guys? .. lol

I mean, I know that especially in statistical classes, you get it beaten into your skull that nothing should be assumed, but am I the only one who just read those questions, immediately thought "1/2" for all of them?

I know I stated earlier that I think its a matter of semantics, but when someone says to me "The set of X where Y" is true, I build the set where you have exactly one of each. perhaps I put too much emphasis on the phrase "The Set" as opposed to "A set" but were it a set with an uneven distribution I would expect a wording like "A set of X where Y is true and have Z distribution" or "A set of X where Y is true plus Z objects".

I.E. to me the set of all integers between 0 and 4 is ALWAYS {0, 1, 2, 3, 4} .. not potentially {0, 0, 1, 1, 2, 3, 4, 4, 4, 4,}

[joft]

Necronus, I think you're assuming that the only probability measure on a given set is the uniform one, and that's not true. Usually any given underlying set has many possible probability measures. In fact, the uniform assumption breaks down for infinite sets, where the only possible uniform assignment to each point is 0.

As an example, consider the set X = { (1/2)^n : n = 1,... infinity }

One possible probability measure on X is the point mass at (1/2) which assigns probability 1 to any event that contains the point (1/2) and probability 0 otherwise. And since sum n = 1 to infinity (1/2)^n = 1, we could also have a probability measure which assigns probability (1/2)^r to the point (1/2)^r for each r

The point is, unless your underlying set is finite and/or you have some reason to believe that the measure is uniform, you can't just assume that it is. And that was also the point of the problem in the OP. The 3 different descriptions of the same set all yield different uniform distributions which then yield different answers to the same question. So not only are there non-uniform distributions, but there are even multiple distinct uniform distributions on the same set depending on what parameter you're using to describe the set.

[Max™]

Overthinking a problem is never bad, you learn more than you will ever do simply taking shit at face value and solving it with the problem, parameters, and everything except the answer taken care of for you.

http://www.youtube.com/watch?v=NWUFjb8w9Ps

Incidentally, am I the only one that thought, when Necronus mentioned "the set of all integers between 0 and 4", 'what about 1+2i? and other Gaussian integers?'... or have I been spending too much time with primes on the brain?