The derivation of the formula is not correct.

If you want to know how large a group needs to be before you can say that there is a better than half chance that at least one group member is engaged in a given activity, assuming that engaging is the activity is independent of other members’ activity statuses, here’s how you would do it. In this example, to make the numbers easy, I am going to use sleeping as the activity and that people sleep eight hours a day and, importantly, the sleeping activity is independent of other group members and is uniformly distributed over the course of a day.

Thus, the probability that a specific person is asleep at any given moment is (8/24)=(1/3). Accordingly, the probability that he is not asleep is (2/3). The probability that everybody in a group of 2 is awake is (2/3)[sup]2[/sup], and in general, the probability that everybody in a group of *n* is awake is (2/3)[sup]*n*[/sup].

The probability that at least one person among *n* is asleep is just that probability that not all *n* people are awake. That is to say, the probability is 1-(2/3)[sup]*n*[/sup].We want to know how large *n* has to be so that

1-(2/3)[sup]*n*[/sup]>(1/2).

Answering this question tells us how large a group we need in order to say that it is more likely than not that at least one of them is asleep.

Reordering this equation gives us

(2/3)[sup]*n*[/sup]<(1/2).

We can find *n* using logarithms.

ln((2/3)[sup]*n*[/sup])<ln(1/2)

*n*ln(2/3)<ln(1/2)

Because the logarithm of a number less than one is negative, we must flip the sign when dividing by it. Doing so yields the following:

n>ln(1/2)/ln(2/3)

In this case, n must be at minimum 1.709511291. Or, since we deal with integer values for human beings, 2.

In general, if *p* is the probability of being engaged in a given activity, then we can deduce from the above the following as the minimum “more-likely-than-not” group size *n*:

*n*=ln(2)/ln(1/(1-p))

In the case of sexual activity, *p* would equal the number of sexual episodes per time-unit multiplied by their average duration in that same time-unit. Using the variables from the webcomic:

*p*=*X[sub]f[/sub]X[sub]d[/sub]*

Once you know this, you can find *n*. Once you know *n*, you can use

*n*=*P[sub]d[/sub]πr*[sup]2[/sup]

to find the minimum radius needed for an estimated *n* people. Making this explicit:

*r*=sqrt(*n*/(*P[sub]d[/sub]π)*).

In particular, notice that the assumption the webcomic makes–that the minimum *n* needed for a better than half probability is (1/2)/(*X[sub]f[/sub]X[sub]d[/sub]*)–is incorrect.