Pascal's Triangle

A number of unsolved problems in Pascal's days encouraged the formation of probability theory. The Gambler's Ruin and the Problem of Point are two examples of such problems.

The Gambler's Ruin was a problem Pascal challenged the great mathematician, Pierre de Fermat, to solve. The problem, according to one explanation, was determining what the chances of winning were for each of two men playing a game with two dice. When an 11 was thrown on the dice by the one man, a point would be scored. When the second man threw a 14 on the dice, he would score a point. The points only counted if the opponent's score was zero. Otherwise, the point scored by one of the men would be subtracted from his opponent's score. So one of the men would always have a score of zero throughout the game. The game was won when one man gained 12 points. Pascal asked, what was the probability of each man winning? Binomial coefficients can be used to answer the question.

The Problem of Points was also a game about probabilities. The question was determining how a game's winnings should be divided if the game was ended prematurely. Questions about games like these stirred the development of probability theory, and the need to understand binomial numbers completely.