Bayes' theorem is a one-line rearrangement — why is it treated as a landmark result?

You are right about the mathematics and that is not what is being celebrated. The importance is what the identity lets you do with the quantities involved, not the identity.

Write it in the form that shows why:

P(hypothesis | data) = P(data | hypothesis) × P(hypothesis) / P(data)

The left side is what you want and cannot observe: how likely is my explanation, given what I saw. The right side is built from things you can get at: how likely was this data if the explanation were true, and how likely did the explanation seem beforehand.

So the trivial rearrangement converts an unanswerable question into an answerable one. That is the entire fuss. It is a bridge between the direction reasoning naturally runs — from cause to observation — and the direction we actually need it to run — from observation back to cause.

No other one-line identity in the course does anything like that.

The second reason is that it makes a genuinely contested philosophical position operational.

The claim that a probability can describe a degree of belief in a hypothesis — rather than a long-run frequency of an event — was and to some extent still is disputed. The theorem is what turns that claim into a procedure: start with a prior belief, observe data, get a posterior belief, repeat. Belief becomes something you can update mechanically and check.

That is why an entire school of statistics is named after it. The name does not mark the difficulty of the algebra; it marks which side of an argument you are on about what probability means. A frequentist and a Bayesian both accept the theorem and disagree about what it is legitimate to put into it.

The popular science attention has a third, more mundane source: it produces results that are reliably counter-intuitive, which makes for good writing.

The standard example is a screening test that is 99% accurate for a condition affecting one person in ten thousand. Most people's instinct is that a positive result means you almost certainly have it. The theorem says it is under one percent likely, because the enormous number of healthy people generates far more false positives than there are true cases.

That gap between intuition and answer is large, the arithmetic fits on a napkin, and the conclusion matters — it applies to medical screening, security alerting, fraud detection and spam filtering. A result that is simple, surprising and consequential is going to get books written about it regardless of how hard the proof was.

Worth noting the part that genuinely is hard, since the theorem itself is not: the denominator.

Computing the probability of the data means summing or integrating over every possible hypothesis, and for any interesting model that is intractable. Essentially all of computational Bayesian statistics — the sampling methods, the approximations, the whole machinery — exists to get around that one term.

So the line is trivial and using it at scale supports a large research field. That is a slightly funny place for a subject to end up, and it is another reason the theorem gets more attention than its proof deserves.