Why the Probability That Two Random Numbers Are Coprime Equals 6/π²

A Surprising Connection Between Random Numbers and π

Imagine you pick two large random numbers a and b. You write down the fraction a/b. Now the question: what is the probability that this fraction is irreducible — that is, that a and b are coprime?

The answer that breaks the pattern: 6/π².

Yes, the number π — the one we know from geometry and circles — unexpectedly shows up in a problem about fractions and integers.

Coprime Probability: Verifying It Yourself

If you want to see it for yourself, try an experiment: take two random numbers, say from 1 to 100 million, and check whether they are coprime. Computer simulations consistently show that the probability approaches 6/π².

This isn’t a coincidence or a numerical quirk — it holds no matter how large you make the range.

How the Riemann Zeta Function Gives Us the Answer

When we talk about the probability that two random numbers are coprime, the key lies in how those numbers “interact” with prime divisors. For each prime, you can calculate the probability that both numbers are not divisible by it — for 2, that’s three out of four possible cases; for 3, it’s eight out of nine; and so on.

Multiplying these probabilities together across all prime numbers leads us directly to the Riemann zeta function. Its value at a specific point — 2 — is what produces our unexpected result: the probability that two numbers are coprime equals 6/π².

Why π Appears Where You Least Expect It

It seems improbable, but π has a way of turning up exactly where you’d least expect to find it. A number born from the geometry of circles quietly governs the arithmetic of divisibility — a reminder that mathematics is far more interconnected than it first appears.