MATH 311-01, Probability Theory, Fall 2009

Abstract

As a ¯eld of mathematics, probability theory is fairly young. Some historians say that the study of probability began in 1654, when Antoine Gombaud and Chevalier de Mere discovered that their intuition about the results of a certain dice game con°icted with their mathematical analysis. They contacted Blaise Pascal and Pierre de Fermat, who began to exchange letters and who jointly formulated some of the fundamental principles of probability theory. Probability theory was not a well-respected ¯eld at the time; it was considered a conglomeration of counting tricks, quirky results, and rough heuristics. Although mathematicians like Jakob Bernoulli, Christian Huygens, and Abraham de Moivre continued to develop the ¯eld, they generally focused on problems associated with gambling. Pierre de Laplace was the ¯rst mathematician to show that probabilistic results could be applied to many di®erent scienti¯c and practical problems, especially in statistics and actuarial mathematics. Probability still remained a messy discipline from a modern mathematical perspective, however. Mathematicians couldn't seem to agree on a good de¯nition of probability that was both rigorous yet still °exible enough to permit the discipline to have broad applications. This matter was ¯nally resolved by the famous Russian mathematician Andrei Kolmogorov in his monograph of 1933, in which he axiomatized the ¯eld. Since that time his ideas have been re¯ned, of course, but due to his e®orts probability theory has now been embedded in a larger ¯eld of mathematics called measure theory. Today, probability theory is central to the study of nearly every applied science, including ¯nancial mathematics, neuroscience, climatology, and quantum mechanics. I hope you will enjoy learning it with me!