Ocean Primality and Log-Fermi

"Could the (rounded whole) number of gallons of water in the oceans be prime?" Though this question from my girlfriend sprang from pure curiosity, it connects well with, and I'll argue improves upon, a long tradition of interview questions.

Interviews for technical positions mythically involve answering questions like "What should you charge to clean every window in NYC?". Such Fermi problems force the recipient to wisely pick intermediate values to approximate and to quickly estimate the relative order of magnitude of related quantities. I hear they're still asked, but I've never been asked a question like this in an interview, even for a quant finance firm.

There are issues with Fermi problems, beyond just the issues inherent to brainteasers as an interview tool. One central gripe is that good answers feel bad. If I ask how many grains of (dry) rice fit in a gallon, and you say 26,000 when the answer is 15,000, it feels far off, even though this is geometrically a better estimate than could reasonably be expected. The point is to roughly estimate the order of magnitude, but when the answer is given in absolute terms, the arithmetic difference dominates our mental assessment of performance.

Then, hanging out one day, a friend asked "How many times do you need to fold a piece of paper to reach the moon?". This type of problem could reasonably be called Log-Fermi. We still estimate a rough order of magnitude, but then we take the logarithm (base 2 in the paper folding case) before reporting a final answer. The logarithm turns the geometric differences into arithmetic differences and estimation errors have a more representative feel. I thought for a few minutes and answered my friend's question "43". The correct answer is 42 folds, and I was so proud of my answer. My original distance estimate was 2× off, but I was within 1 of the correct answer. Loosely, Log-Fermi is more fun and more humane.

Months later, I was asked the original question by my girlfriend: "Could the (rounded whole) number of gallons of water in the oceans be prime?". Our gut reactions were "almost certainly not"; the primes thin out further down the naturals. Heck, there's already a 50% chance it's just a multiple of 2! There are about 3.53 × 10²⁰ gallons of water in all oceans (source). In a Cramér-style model, there's a 2.66 × 10⁻¹¹ probability that this number is a square; lower probability than winning the lottery.

But primes are significantly more common than squares; a number n is prime in the Cramér model with probability 1/ln(n). We use this to calculate that, at any given moment, the probability that the number of gallons of water in the ocean is prime is about 1/47.3 or 2.1%; this is higher than I expected! In fact, taking into account the nonindependence of primes in an interval, if the number of gallons in the oceans fluctuates by >550 gallons (a hot tub) in some time interval, then there's a >99.999% chance it takes on a prime number of gallons within that time.

While I was surprised by the high likelihood of ocean primality specifically, the point isn't the magnitude; it's that the Prime Number Theorem gives us another approach to Log-Fermi, one that is not length/distance-specific like folding a piece of paper. We could easily use it as a frame for problems about grains of sand, metro riders in a city, or stars in the observable universe.

This post has two natural paths to Log-Fermi problems; are there more? It's a fun question I'll continue to mull over.