Interaction between Melchior and me at reddit/math has got too long to continue in its original form. Here’s what’s gone before in response to my recent article about the Bertrand Paradox (note that by some cruel twist of fate, the name neopolitan was not available to me):
You pose an ill-defined problem to which there is no right or wrong answer, then you nominate one answer as "probably right" for dubious reasons. You stick with the vague "select at random from a set" concept and try to work with subset relationships, but this approach can't express the hidden assumptions that underlie the paradox.
I posed the question as "what is the probability, p, of a random chord in a circle being longer than the side of the largest equilateral triangle that can be drawn in that same circle?" Wikipedia has it expressed as "Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?"
I don't really think that either question is ill-defined.
It only appears ill-defined when you select chords from a biased set (thus making them not random).
My set was only biased so as to make sure the two random points result in a chord, I don't think that that is invalid, is it?
But, yes, I agree, I do have to make that a little more clear in the original article. Thanks again.
| “It only appears ill-defined when you select chords from a biased set”
There's no such thing as a "biased set". Every probability distribution is "biased" relative to some other probability distribution.
| “(thus making them not random)”
The phrase "not random" is meaningless here. I think you mean that the distribution is not uniform, but that's also meaningless unless coordinates are specified, and it does depend on the coordinates.
Ok, maybe this is why I should have paid a little more attention to statistics many years ago.
Re a “biased set” – if you are considering the probability of selecting a human over 180cm tall and you restrict your measurements to all American males, then you are not going to get the correct result for all humans. This is what I mean by a “biased set”. You might want to call this a “biased sample”, but note that I said ALL American males. If you, for some reason have got it in your head that the set of all humans is identical to the set of all American males, then you have a biased set.
This is similar to the case with the p=1/3 calculation of the probability of randomly selecting a chord of length greater than r.√3 in a circle of radius r. If you restrict your selection of points which define a line which then results in a chord to any two on the circumference of the circle, you get a “biased set”. The biased set results in p=1/3 while an unbiased set (which I think I’ve defined) results in p=1/2. I’m sorry if my terminology is not standard but if you can try to see past that it would be much appreciated.
Re “not random”, hopefully this is clarified by the above but to make sure, by this I mean that a random selection from the set of all possible chords is not going to give you the same result as the random selection of a pair of points on a circle, even if those points can then be used to define chords. A “random” selection from a subset of all chords is not necessarily equivalent to a random selection from the set of all chords. (It’s like trying to select people at random from a crowd and subconsciously only selecting females that you think are particularly good-looking, it might be a random selection of good-looking females, but it’s not random in terms of the task at hand.) Again, I might not have used the correct terminology for which I apologise.
By the way, are you a p=1/3 person? Or simply undecided?
The set of humans is simply different from the set of American males. Neither one is biased.
The distinction is a red herring anyway, because we're talking about one and only one set: the set of chords in the circle. You keep saying "subset", but the set of chords defined by choosing two endpoints is precisely the same set as the set of all chords. There is no chord that exists in one set but not the other.
Subsets can't rescue you from this paradox. It's not just a matter of terminology; there's something very different going on here. This is a really important point, and you have to accept it in order to understand anything else about the problem.
This last response is quite interesting.
Wikipedia defines a paradox broadly enough to include puzzling results that can be rectified by demonstrating that one or more of the premises themselves are not really true, are a play on words, are faulty and/or cannot all be true together, going far enough to state that paradox is often used interchangeably with contradiction. Even dictionaries seem to be rather vague on the topic, for example the first entry at merriam-webster.com isn’t even remotely close to what I personally would define as a paradox.
Let me clarify. In this context I mean by paradox, as I am sure the people who originally named it the Bertrand Paradox meant, a problem which appears to have (at least) three alternative solutions which cannot all be correct. It cannot be true that p=1/2 and p=1/3 and p=1/4 in this scenario, when p is the probability that a chord selected at random is greater than r.√3.
I know that Bertrand originally presented the problem as an example of how different methods of producing a random variable can affect the result, but I think you might be interpreting this incorrectly. A truly random selection of a chord is not necessarily equivalent to following a process which is (in a sense) random and which results in a chord. You might think that, by following each of the three methods listed in the Wikipedia article, you are randomly selecting a chord, but my argument is that you are not. You are close to it with method 2 (the radius method), but you are not with methods 1 or 3 – unless you modify them in the way that I have described in my earlier article.
I am curious as to your intent, Melchior. It may be a tiny bit unfair to use this medium to analyse your responses, but life is always a tiny bit unfair, so I’ll carry on regardless.
You initially said the problem is ill-defined. It’s not. You also made mention of the “hidden assumptions that underlie the paradox”, but you’ve not made clear what these hidden assumptions are. Accepted that I haven’t previously asked you to, but I am now: What are the hidden assumptions underlying the Bertrand Paradox?
Then you totally misunderstood the point about the “biased set”. The set of American males is a set and the set of humans is a set, it’s generally accepted that the former is a subset of the latter. The problem arises when you take only American males and assume that the results of any measurement will be representative of humanity as a whole. The distribution of chords as generated via two random points on the circumference of a circle is not the same as the distribution of chords as generated by two random points selected without the constraint. The former is a subset of the latter. Do you not agree?
I did ask you if you were a p=1/3 person or just undecided, which you didn’t respond to. Could you please be open about that?
Finally, you say that subsets “can’t rescue (me) from the paradox” and “there’s something very different going on here”. Then you talk about understanding “anything else about the problem”.
When you talk about rescue from a paradox, it seems to me that you are talking about inherent self-contradiction as in the following couplet:
The sentence below is a lie.
The sentence above is true.
That’s an inescapable paradox caused by indirect self-reference, you have to just walk away from it rather than try to work out whether one or both or neither of them are correct.
Are you implying that the Bertrand Paradox is a true paradox in the stricter sense? Is the “something very different” that you are hinting at merely a reference to the problems associated with simulating randomness (which Bertrand was in essence talking about)?
And, as a last question, what is the “anything else about the problem” of which you speak?
Sincerely in anticipation of a fruitful continuation of this discussion,