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The Evolution of Probability: Degree of Belief

The Problem of Points existed for a long time before Pascal and Fermat solved it. Why didn’t they solve it sooner? The short answer is: degree of belief.

A simple definition for “degree of belief” is somewhat elusive. And it gets trickier when we look at the historical context of the idea. One theory for why Pascal and Fermat solved the Problem of Points is that no comparable mind existed before them. A more complete answer is that, until the Enlightenment, little attention was paid to probability. In the deterministic universe of earlier thinkers, chance was the domain of gods or God. It was not man’s place to attempt to predict the future. In the changing economic landscape of the 17th Century, though, there was an increased need for a new kind of mathematics.

Pascal and Fermat’s solution of the Problem of Points led to a definition of modern probability:

The calculation of the relative frequencies of outcomes of interest within the universe of all possible outcomes.

This view of probability has led to its propagation as a useful mathematical tool. This mathematical view has slowly led to a deeper understanding of what probability means, and what powerful ways it can be used. But it took most of the lifespan of humanity for us to reach this point. And it has taken about four hundred years to gradually percolate into multiple fields of study. Again, the question arises: why so slowly?

“Degree of Belief”

Pascal and Fermat’s work on the Problem of Points meant something more than just a statistical view of probability. And it wasn’t just a new brand of mathematics, like calculus. One of the breakthroughs in probability was as much philological as it was mathematical. In an early foreshadowing of this mini-series, we briefly touched on this philological evolution. The immense impact of this revolution in thought cannot be overstated, however.

For much of human history, the word “probability” did not have the modern meaning we’re all used to. Its meaning was closely attached to words like “probably”, and to the belief in a higher power. For this reason, probability was not only considered imprecise, but also unquantifiable. It was beyond the scope of man’s mind, so the thinking went. Probability represented the “known unknowns” and “unknown unknowns” of life.

Modern probability is attributable to Pascal and Fermat, but equally so to their historical era. In modern times, we take for granted the separation between mathematical probability and what we might call logical, or epistemic, probability. Before the 17th Century, our degree of belief in something unquantifiable was not considered worthy of further study. There was thus no distinction between everyday probability and its mathematical alter-ego. This is the form of probability on which many gamblers rely. It also can be used to describe the simple decisions we make every day. Our gut feelings. Our subjective beliefs. The rational, but not mathematically defined, forecast of what we assume will happen.


The Renaissance and the subsequent Enlightenment were based on the pursuit of certainty. The major thinkers of this era, Pascal and Fermat included, sought certainty where before there had been only vagueness. Gods and kings and man’s gut instincts were not enough for these men. The new sciences – physics, astronomy, modern medicine, et cetera – had to have definitive, actual proof of their conclusions. This led to a fundamental change in the way humans think. In this way, the solution of the Problem of Points helped to usher in our modern era of scientific understanding.

This is Part Two of a series on probability. Click here for Part One, Part Three, Part Four, and Part Five.