Math (probability) puzzle

[CDF]

If the question is not well-defined, I'm going with the smart-ass frequentist answer.

If you aren't actually trolling, you really need to know the distribution of one of side length, area, or volume to answer the question. Knowing the distribution of one determines the distributions of the others as we are talking about a cube.

Suppose side length is uniformly distributed. Then all three probabilities are 1/2.

Suppose area is uniformly distributed. Then all three probabilities are 1/4.

Suppose volume is uniformly distributed. Then all three probabilities are 1/8.

[Max™]

http://ants.planet.ee/trap/boxxy-trolling.gif

[Maxxthepenguin]

If the answer is so trivial, Mr. penguin, why not give a clear explanation for those of us who aren't as brilliant as you? Why do I want to answer 1/2, 1/4, and 1/8 to the same question stated differently?

Assuming the distribution is uniform between 0 and 1 for a side (an equal probability for every number between 0 and 1) then you have a 50% chance to get .5 or less for a side (obviously). 1/2^2 calculates the area of a side, so that same 50% chance applies to an area between 0 and 1/4. 1/2^3 calculates the volume so that same 50% chance applies to the volume between 0 and 1/8.

There. I just wasted my time for you. I hope you're happy.

[joft]

Sorry Mr. penguin, but I liked CDF's explanation better. CDF wins the thread

Each rephrasing of the question is intended to tempt the reader into applying a uniform distribution on a different measurement, size, area, or volume. In the absence of any information or evidence about the distribution, it makes sense to assume the distribution is uniform (this is sometimes called the principle of indifference). The question mentions 3 different parameters that each completely describe the set, but we have to pick one of them to be distributed uniformly. Once that decision is made, the answer is fixed.

[Maxxthepenguin]

So it's a riddle that everyone will always get right, even if you don't explain it how you'd like it.

[The_OG_Nelta]

http://ants.planet.ee/trap/boxxy-trolling.gif

http://i40.photobucket.com/albums/e2...fdi97r6q5y.gif

[Max™]

Can you apply the principle of indifference to a frequency?

That is something which was also fucking with me, kept trying to pick out a reason not to select all three descriptions.

[CDF]

I'm pretty sure that whatever distribution one of them is, you'll get the same cumulative probability. It is a cube after all. You would just have to figure out the cumulative probability for one of side length, face area, or volume.

[Galois]

1) 1 or 0
2) 1 or 0
3) 1 or 0
4) The cube was selected already.

This is the correct answer from a frequentist standpoint.

[Max™]

Yeah, but isn't indifference a Bayesian thing?

[joft]

Mr. penguin: The challenge was to explain what was wrong with the question. Communicating clearly and precisely in the context of an ambiguous problem is one of the most important skills any quantitatively oriented person needs to have.

Max(tm): I don't know what you mean by applying it to a frequency.

Galois: Okay funny guy, then let's say we pick a few thousand random cubes. Approximately what do you think the proportion will be?

[Max™]

http://mathworld.wolfram.com/BayesianAnalysis.html

http://mathworld.wolfram.com/Probability.html

Principle of Indifference is a Bayesian thing, but the frequency you're asking for sounds like it is phrased in a Frequentist manner.

[joft]

I still don't know what you're talking about. Were you replying to my comment to Galois? Other than that, I never asked for a frequency.

BTW, strict frequentist philosophies don't even apply to this problem. It's impossible to ever perform the experiment of picking a random real number between 0 and 1.

[Max™]

Uh... perhaps I have a different definition of 0, 1, and real number than you do?

[joft]

I'm certain that's not the case.

[CDF]

He did not mean principle of indifference in Bayesian terms, but more along the lines of assuming a distribution that happens to be uniform (because why do something annoying like assume a beta distribution), at which point you are either right or wrong about the assumption. So that is more like a principle of convenience.

[Galois]

I was posting in terms of that (in the frequentist paradigm), once the cube is selected, the lengths of its sides is no more random than the number of faces it has; either it has a length in a given interval or it doesn't.

edit: CDF beat me to it.

[Max™]

Uh, there are an infinite number of real numbers (and thus possible cube lengths) between 0 and 1, frequentist methods involve seeing if a result is possible, Bayesian methods involve seeing if a result is believed to be possible.

Bayes allows one to take a frequency prediction after formulating it in that manner, but a frequentist interpretation is simply if it did or did not happen, true or false.

If you are treating it that way, then the Principle of Indifference is at least oddly applied, since you are tacking on the assumption of Bayesian principles, which was kinda fucking with me due to your wording.

[Galois]

I was only considering the interpretation of probability under the two schools of thought; I wasn't speaking towards assigning prior distributions.

Edit: Sorry for derailing things

[joft]

Max, if you had countably infinitely many computers, each capable of choosing countably infinitely many real numbers between 0 and 1 every Planck time unit, and you could wait for countably infinitely many years, they would still only choose a countable subset of the interval. Countable sets have zero Lebesgue measure.

You can choose a real number, but you're essentially infinitely biased against almost all of them.