There is no way on this planet or in this solar system or this universe I could have been more clear: I responded to

> ... a frighteningly small sample to draw conclusions from?

That's all I responded to.

No way did I try to answer all possible probability and statistical questions in astronomy and astrophysics back to the Big Bang, hope for a Nobel Prize, apply for a chaired professorship, .... Again, I just responded to

> ... a frighteningly small sample to draw conclusions from?

In really simple terms, with common assumptions, the accuracy of a sample mean is JUST from the size of the sample and has nothing to do with what fraction the sample is of some population the sample was drawn from.

This point seems to be worth making. E.g., once while working myself and my wife through our Ph.D. degrees, I was asked to estimate the survivability of the US SSBN fleet under a special scenario. From some WWII era Koopman work on encounter rates, and more, I found a continuous time, discrete state space Markov process and generated sample paths via Monte Carlo techniques. Actually used the random number generator that passed the Fourier test and was published by a group from Oak Ridge and based on the recurrence

X(n+1) = X(n)*5^15 + 1 mod 2^47

So I wrote some code, generated sample paths, and found event by event, essentially hour by hour, the expected number of SSBNs surviving. Right, the scenario included lots of weapon types so had lots of combinations of what was surviving.

It happened, nearly uniquely, that my work right away got a little review from a well known probabilist. He asked:

"How can your scenario generation fathom the enormous state space?"

that is, the number of combinations.

This question is related to

> ... a frighteningly small sample to draw conclusions from?

Soooo, even a well qualified probabilist can ask such a question.

I answered: Pick a point in time, t. There the number of SSBNs surviving is a random variable. It is bounded. The sample paths are independent. So, each sample path gives an independent random variable for time t. So we have an i.i.d. case. So the strong law of large numbers applies. So, run off 500 sample paths, add them up, divide by 500, and get a good estimate of the number of SSBNs surviving within a gnat's ass nearly all the time. The probabilist was socially shocked and offended by my mention of a "gnat's ass".

I went on, intuitively, Monte Carlo "puts the effort where the action is". The probabilist's response was "That's a good way to look at it."

In a sense, the most common sampling situation is a finite sample from an infinite population so that the sample is 0% of the whole population -- and we are back to

> ... a frighteningly small sample to draw conclusions from?

0% "small". So, actually 0% small is common.

So, it is common for people to consider what fraction the sample is of the whole population being sampled. This being the case, in my post here I continued and outlined some ways regard the population as finite. That can take us into sampling from finite populations which can be conceptually tricky -- I want to avoid that stuff, and for something the size of the universe, maybe not for a deck of 52 cards, that is reasonable.

For a simple observation, put 50 independent random variables in a box and they are still independent.

That's all folks!