Why Probability Theory Is the Mathematics of Randomness

Summer is the perfect time for unusual mathematics. During the school year, everything follows a strict plan — what to study, which exams to take. But summer always brought a small sense of freedom. Two months to think about other things. Sometimes even about something genuinely fascinating.

What Makes Probability Theory Different from Other Mathematics

For example, probability theory. If you stop to think about it, this branch of mathematics is strikingly different from everything else.

Consider the contrast: mathematics typically describes shapes, objects, and the relationships between them — both in the real world and in the abstract. Then suddenly it turns out that mathematics can describe randomness itself.

It’s not at all obvious why things that either happen or don’t happen can be tied to strict mathematical rules.

Why Randomness Belongs to Probability Theory, Not “Chancology”

Physics describes the laws of nature. Mathematics describes patterns. Biology covers living organisms. Chemistry covers substances. At first glance, it seems surprising that mathematics ended up describing randomness — and not some invented field we might call “chancology.”

Although, on the other hand, random events turned out to be not so random after all — just a collection of patterns in disguise.

The Hidden Patterns Inside Probability and Statistics

What’s more, there aren’t actually that many of these patterns. Around ten key distributions cover most cases, and completely different phenomena end up being described by surprisingly similar processes. Ordinary dice and playing cards, for instance, reduce to the simplest fractions.

The world around us is often chaotic and uncertain. And here comes a tool that allows us to describe that uncertainty — probability theory.

How Probability Theory Makes Uncertainty Manageable

Admittedly, for any individual case or person, this description doesn’t offer much comfort. The world remains just as chaotic and unpredictable as before.

But for all people taken together, or viewed over a long enough horizon, uncertainty becomes quite manageable.

Reading Feller: Where Every Uncertain Problem Has a Definite Answer

All of this makes a strong case for picking up Feller and working through some problems. At the very least, there every question about uncertainty comes with a definite answer.