Borel-Cantelli lemma and the Universe

November 6, 2021


A science-fiction theme that never goes out of fashion. From friendly neighborhood aliens to blood-thirsty villains determined to enslave the earth, we have captured them all in our imagination.

What makes this theme so enticing? I vividly remember wondering this out loud in frustration one day. It was on a night during a tech-fest week at our college. I had just finished imparting the basics of star-gazing to a small gathering at the astronomy workshop that I helped organize and asked if they had any questions.

“So what do you think are the chances of having any aliens out there?” I took just long enough to roll my eyes hoping the darkness would cover for me and answered, “No one knows, do they? Now let’s just get back to the stuff that I talked about telescopes and messier objects.”

Every single time!

I guess that speaks to our social natures. We do crave company. Seven billion people on the planet and we still can’t stop thinking about whether there’s someone else out there that we could reach, talk to, or even fight.

So let’s talk odds. Not just in the figurative sense but in the literal sense. Frank Drake, an astronomer, set out to note down the probability of extraterrestrial communication as a thought experiment and gave us the legendary Drake equation,

N = R * Fp * Ne * Fl * Fi * Fc * L

which counts the number of extraterrestrial civilizations in our galaxy that we are likely to hear from.

The equation is a probabilistic argument with a gradual winnowing of events that starts with the rate of average star formation in the galaxy (R).

How many of these stars are likely to have planets (Fp)?

Of these, how many are likely to support life (Ne)?

How many actually go on to develop life (Fl)?

Among them, how many are likely to result in intelligence (Fi)?

And the likelihood that the intelligence is strong enough to develop technology capable of communication (Fc)?

Lastly how long do these civilizations actually transmit signals into space (L)?

A purely speculative argument with factors that are incredibly difficult to compute. But still, it puts forth an interesting idea that the evolution of intelligent life is a probabilistic event.

In my Probability and Algorithms class at graduate school a week ago, I was reminded of the Drake equation. It was during a lecture about the Borel-Cantelli lemma, a significant result in measure theory. The lemma, as most math theorems do, sounds very obvious but does need rigorous and not-so-trivial proof for establishing its validity.

The first Borel-Cantelli lemma says that given a sequence of events in some probability space, If the sum of the probabilities of the events is finite, then the probability that infinitely many of them occur is zero.

Sounds obvious, right? The second Borel-Canteli lemma, a partial converse of the first is far more exciting. It says,

If a sequence of events in some probability space are independent and the sum of their probabilities diverges to infinity, then the probability that infinitely many of them occur is 1.

An interesting consequence of this lemma is the infinite monkey theorem, which is a product of the metaphorical minds of scientists. It states that a monkey hitting typewriter keys at random for an infinite amount of time will almost surely type the complete works of Shakespeare.

This result is easy to prove. If you assume that Shakespeare’s text begins at the time t with a probability p (an event Et which is extremely small but still non-zero). All the events Et at different times t are independent and their sum over an infinite t goes to infinite (equivalent to adding the same number an infinite number of times). So the probability that the event occurs infinitely often is 1. So it is not only that the monkey would be able to type the complete works of Shakespeare, but it can do so infinitely often.

The infinite monkey theorem has been popular among atheists and was evoked by Richard Dawkins in his book The Blind Watchmaker. It is an argument against the creationist view of the universe which maintains that intelligent life is intricate enough to warrant a creator and cannot arise out of random chance (hello, infinite monkey theorem?). Given infinite time and space, the event that intelligent life develops is an inevitability, argues the Borel-Cantelli lemma.

The lemma not only says that the event of intelligent life is inevitable but it also occurs infinitely often. Sounds promising! We are an embodiment and a witness to a rare event in the universe but by no means singular, if math is to be believed.

So what’s the chance of meeting one of the other infinite possible life forms? Very slim. Almost as slim as of the monkey typing two complete works of Shakespeare in a row. True, there might be infinite such life forms but the likelihood of two such events occurring at temporal and spatial proximity is extremely unlikely. So, I don’t see an extraterrestrial making a hyperspace jump and contacting us anytime soon. Maybe I’m just not a believer.

But of course, the whole argument hinges on a pretty big caveat. Our primary assumption that the evolution of intelligent life is a probabilistic event that is prone to recurrence can easily be challenged. What if life’s just a one-off anomaly that can never be replicated? Can we think of any such event that has happened once and we can say with absolute certainty will not happen again? I’m blanking here.

I’m a huge believer in a probabilistic view of the universe and that we are here as a consequence of a sequence of random Shakespearean events and mutations. And as Borel-Cantelli tells us, it will happen again given an infinite-time universe. I might never get to meet them or know of their existence but I do hope it goes well for them as I hope the same for us.