THE THEORY OF PROBABILITY1.2 THE DEFINITION OF PROBABILITY1Chapter 1The Theory of Probability1.1STATISTICAL MECHANICS AND PROBABILITYStatistical mechanics is the theoretical bridge between the microscopic world and the macroscopic world.The microscopic world is a quantum mechanical world, in which electrons, protons, and neutrons organizethemselves into atoms, molecules, crystals, and other structures on an atomic length scale. The macroscopicworld is the tangible world of trees and rocks and water. The microscopic structure of these things is usuallycompletely hidden from our view. It is only by a complex chain of reasoning that the fundamental laws ofphysics, which govern the structure and interactions of the unseen microscopic constituents of things, canbe shown to give rise to the observed physical properties of ordinary objects. That chain of reasoning is thesubject of statistical mechanics.A typical macroscopic object, a glass of water, contains about 3×1025molecules. The extreme com-pactness of exponential notation masks the vastness of that number.If that glass of water were mixedthroughout all the oceans of the world, then a glass of water, drawn anywhere on earth, would contain manyof the molecules that had been in it. Clearly, there is no possibility of ever knowing the exact microscopicstate of a glass of water.In statistical mechanics we will have to work with very incomplete informationabout the systems we are analyzing. Typically, we can express our information about the state of the sys-tem in 3 or 4 numbers, such as the system’s volume, mass, temperature, and pressure, while a completespecification would need about 1026separate numbers. It is remarkable that, with this meager information,we can predict so many important things with high precision and reliability. For other things we can makeonly probabilistic predictions. But, even for those, the methods of statistical mechanics permit us to makeaccurate predictions of the probabilities.The general mathematical discipline involved in making predictions with limited information is proba-bility theory. In a real sense, statistical mechanics is a branch of probability theory. However, the systemsunder analysis in statistical mechanics have certain special characteristics that have allowed the subjectto develop a subtlety and predictive power that are unmatched by any other subject in which statisticalmethods are used. This chapter will be devoted to an introduction to the general ideas of probability theory.All the following chapters will specialize the theory to the problem of calculating the physical properties ofaggregate matter.1.2THE DEFINITION OF PROBABILITYThe reader certainly has some idea of what probability means and most likely knows how to make simplecalculations of probabilities. In order to construct a mathematical theory of probability, we will first look ata problem in probability that is easy to solve and then abstract the rules we use to solve that problem and