There’s a link at the top of the page: [Example Google sex calculation](http://www.google.com/search?hl=en&q=sqrt(2/(pi*18000/(mi^2)80/year30+minutes)).
Rather than working out the minimum group size to make a single copulator more probable than not, though, isn’t it simpler to do it as a straight density calculation? In a given square mile containing P[sub]d[/sub] people, we know that some proportion X[sub]f[/sub]X[sub]d[/sub] are having sex at any one time. We can assume that they must have paired off and co-located themselves in order to achieve this (we’ll ignore threesomes and the exceedingly well-endowed), effectively halving their density. So we can simply say that there are .5X[sub]f[/sub]X[sub]d[/sub]*P[sub]d[/sub] copulating couples within each square mile, which, when you solve for the minimum radius required to contain one such couple on average, gives precisely the result in the comic.
This seems to obviate the dependency problem completely, and I can’t see any logical missteps.
The difference is between expected value, which is what you’ve calculated, and my value, which determines the group size at which it is more probable than not that someone is engaged in sex. To illustrate, pick some day of the year, say June 1. What group size do you need to make it more probable than not that someone in that group has a birthday on June 1? The answer is ln(1/2)/ln(364/365) ≈ 253. But if you asked, how many June 1 birthdays can we expect in a group of 253? That answer is (253/365). This is because although it is more probable that not that someone has a June 1 birthday in that group, it is also possible that none do, or two do, or three, or even all 253 do. The expected value of all these possible outcomes, the sum of the probability of each possible number of June 1 birthdays times that number or Σ(i=0, 253, C(253,i)(1/365)[sup]i[/sup](364/365)[sup]253-i[/sup]*i), is precisely (253/365).
I suppose one fair understanding of the question “What is the minimum radius for which we can say there is probably sex going on?” could be, in more precise terminology, “For what radius would we expect the number of people then engaged in sexual activity to equal two?” The question would be answered by your method. This could even be expressed as “the radius for which, on average, the number of couples currently having sex within that radius is one.” This might be the point Chronos was urging above.
My method provides an answer to the question “At what radius can it be said that it is more likely than not that someone is engaged in sexual activity?” It will be smaller than the one your method produces and I think that mine is the notion that people try to capture when they talk about radii of probable intercourse.
This is precisely what I was going for, and I think it’s arguably the most natural reading of the comic’s phrasing. At any given time, the average distance to a pair of filthy copulators would be this, which to my mind at least is what’s at issue (as it were). The comic did, after all, specify average distance rather than 50% probability.
I do see your point, though; it just makes the calculation far more complex, and introduces an assumption (independence) that isn’t really supportable and underestimates the radius, since it implies an even spatial distribution of copulators, when in fact they have to be clustered in groups of n>=2. In particular, with reference to your first post, it seems unfair to say that the comic makes an incorrect assumption re: probability. To my mind, it doesn’t; it’s calculating a different value based on perfectly fair assumptions.
Well, you make two mistakes here. First, you EV radius does not determine the average distance to a pair having sex. Second, I’m not sure your method does properly account for the fact that when we deal with couples having sex, we no longer have independent events. In order to do so, you would need to introduce a term for the covariance.
I don’t think that the non-independence due to it taking two to tango is significant in practice. You could, instead, consider just the population density of males, and then consider the characteristic distance (whatever definition you use for that) to a male having sex. Or the same thing for females. Since most places have essentially the same population density for males and females, it should work out the same.
I would certainly have thought it would be less significant than the error due to an implicit assumption that sex-having people are evenly distributed, which will surely lead to a fairly hefty underestimation of the radius.
I don’t think I was clear (in fact reading again “at any given time” was completely not what I meant); the EV would be the average measurement of a repeated series of tests measuring the distance to the nearest couple having sex (right?). This, for me, is the most natural reading of the comic’s phrasing, but I happily admit this is entirely subjective.