The Evolution of Probability: Problem of Points

Our Intro series revolves around concepts related to machine learning, but this is a modern invention. The origins of machine learning lie centuries in the past. Blaise Pascal and Pierre de Fermat pioneered the train of thought that would one day give us machine learning. A while back, we touched briefly on these men, the history of probability, and its relationship to legal data. Pascal and Fermat were the first innovators in a brand new field: probability. Today we dig deeper into that history.

The Pascal-Fermat Correspondence

In 1654, Blaise Pascal and Pierre de Fermat began a correspondence on a range of topics. One of the topics they discussed was gambling. Up until that moment in time, only gamblers had cared much for the analysis of gambling problems. But Pascal and Fermat sought to apply more robust mathematics to the questions posed by games of chance. They wanted to discover the best and most accurate way to solve a simple gambling problem known as the Problem of Points. Their solutions to this problem would open up new doors in mathematics and other fields. The invention of probability theory co-evolved with other Enlightenment ideas, such as “proofs” of God’s existence, and the logical assessment of testimony in trials. These new approaches to the sciences and humanities began a slow and incremental revolution. That revolution led to profound changes in human endeavor the effects of which are still felt today.

The Problem of Points

The problem Pascal and Fermat set out to solve seems deceptively easy. Two players participate in a game of chance. They are both equally skilled, and have equal chances of winning any given round. The players both contribute equally to a prize pot. The players also agree to a set number of rounds one must win to be declared the victor.

The next part of the problem concerns what should happen if the game is interrupted and cannot continue. The pot would need to be split fairly based on each player’s probability of winning the remainder of the interrupted game. An earlier solution to this problem divided the stakes in proportion to the number of rounds each player won. Pascal and Fermat were unsatisfied with this solution. The number of rounds already played was unimportant compared to number of rounds left before one player won. The solution could not be based on the past, essentially. It had to be based on all possible futures.

Fermat reasoned the following:

– If r is the number of rounds Player One needs to win, and

s is the number of rounds Player Two needs to win, then

– the game will be over after r + s – 1 rounds

Pascal further refined the formula into the following: 2 ^ r + s – 1. This represents the list of all possible future outcomes. A fair division of the pot consists of the proportion of outcomes where r wins versus the proportion where s wins. Pascal improved this model yet further with a summation formula. The calculation of all probabilities for the Problem of Points is thus expressed by the following:


This relatively simple concept has snowballed over the centuries. Next week, we will discuss exactly why that happened, and how probability has reshaped and refined numerous disciplines.

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