Lateral Thinking

I hate the Drake equation. Actually, maybe that's a little harsh. As I understand it, the intent was never to actually calculate how many intelligent alien cultures there are bumbling around our galaxy, but more to ascertain exactly what unknowns would be required to work such a thing out. Since several of these unknowns are pretty much massively fucking unknowable, though, the Drake equation essentially consists of someone very, very smart shrugging their shoulders. I really don't need to see the working out to know we're not in a position to work out the expected number of sentient extra-terrestrial species, thanks.

There are other things that can be considered instead. Take this idea, for example. Since every star system can be more or less considered independent of every other (or so Pause tells me, though he did remind me about panspermia), there are a finite number of them, and every system either will or won't include "life" (however one chooses to define it), we already have three of the four conditions necessary to model the number of inhabited star systems by a binomial distribution.

The fourth condition, that the probability of a system supporting life is the same no matter what system you are considering, is where we get into trouble. Firstly, and this is where the Drake equation got scuppered, we can't possibly calculate that probability at this stage in our scientific advancement. The variables are too many and too complex, and we only have one data point (us) with which to work with.

This is why we need to work backwards. Instead of trying to (more or less) arbitrarily assign a value to this probability, why not use what we do know (the number of stellar systems in the Milky Way) to work out what that probability would have to be before we would be more likely to be alone than not.

The other problem with this approach is that it's patently ridiculous to assume that each system has an equal chance of supporting life. We'll just have to muddle through with some nebulous concept of "average" probability, and hope no-one asks awkward questions.

I am reliably informed (by Pause, again) that there are something like 100 000 000 000 stellar systems in our galaxy. We'll call our unknown probability p. We're thus considering X~Bin(100 000 000 000,p). Substituting that into the the binomial equation tells us that, for a given p, the probability that there is more than one stellar system supporting life (we already know there's at least one) is:

1-(1+99999999999p)(1-p)^99999999999

In other words, p has to be less than 0.0000000000169 before it's more likely than not that we're alone out there.

This, of course, isn't the full story. Technically we should be considering the conditional probability that there is at least two life-bearing systems given that there is at least one. That gives a new bound for p; now it has to be less than 0.0000000000126 before the odds are we're all alone in the night. [1]

That's about five thousand times less likely than winning the lottery on your first ticket. Depressingly, that doesn't really seem too outrageously low, considering all the factors involved. I'm not really one for biology, though, so I suggest you draw your own conclusions.

Update: Pause points out in comments that xkcd isn't a big fan of the Drake equation either.

[1] It has to be said that things get more complicated when we realise that the only reason we can perform this analysis in the first place is because we're already here, i.e. we ourselves are a product of the observed data. Frankly, though, it's too much of a headache to go into the implications of that, at least for now.