The Fermi Paradox Just from Rarity

1 hour ago 1

By John Mount on November 22, 2025 •

Sabine Hossenfelder’s video “Odds Are The Universe is Full Of Intelligent Life, Mathematician Finds” recently pointed out a new paper on the Fermi paradox: Antal Veres “The solitude zone: A probabilistic window for singular lifeform existence”, Acta Astronautica Volume 238, Part A, January 2026, Pages 1137-1145.

Initially I was a bit put off by the paper’s seeming avoidance of conditional probability. And, as you will see, I emphasize a different summary and conclusion than Dr. Hossenfelder. The paper defines the random variable X as the number of lifeforms (of interest) given N potentially life-supporting systems (also assuming at most one per system and independence of systems). The paper then establishes claims such as:

Pr[X = 1] > Pr[X ≥ 2].

I was wondering: what if we moved more quickly to Pr[X = 1 | X ≥ 1] or the probability there is exactly one interesting species given there is at least one? We look at the conditional probability using the following. We know there is at least one species of interest: ourselves.

Let’s use the paper’s binomial distribution framework with N (a large integer input) trials and p probability (to be determined later).

Define f(p) = Pr[X = 1 | X ≥ 1].

Then:

f(p) = Pr[X = 1 | X ≥ 1] = Pr[(X = 1) and (X ≥ 1)] / Pr[X ≥ 1] = Pr[X = 1] / Pr[X ≥ 1] = Pr[X = 1] / (1 - Pr[X = 0]) = (N choose 1) p^1 (1-p)^(N - 1) / (1 - (N choose 0) p^0 (1-p)^(N - 0)) = N p (1-p)^(N - 1) / (1 - (1-p)^N)

This in fact repeats the point of the paper.

We can derive lim_p->0 f(p) = 1 and f(p) ~ 1 - (N - 1) p / 2 for small positive p (say p << 1/N). This means: if interesting species are a very rare event then even in a universe that has one interesting species, there probably are no others!

So why is that surprising to me? It is in fact what the paper claimed. I think my intuition was wrongly centered around p ~ 1/N. This is from the (incorrect and fallacious) assumption that if we see at least one success in N trials than the success probability is probably at least 1/N. This is false as it completely ignores any sort of prior distribution for p! And priors are in fact one of the points of the paper.

What I take from the paper is the following. What if p is much less than 1/N, and we just happen to be lucky to be here? Then we probably are alone. Just from probabilities, no great filter or silent forest required. This isn’t new, it is essentially “the rare Earth” hypothesis amplified (hence the paper’s discussion of complexity of lifeforms to justify low probabilities).

Categories: Expository Writing Mathematics Opinion