In preparation for our study of statistical thermodynamics, we first review some fundamental notions of probability theory, with a special focus on those statistical concepts relevant to atomic and molecular systems. Depending on your background, you might be able to scan quickly Sections 2.1–2.3, but you should pay careful attention to Sections 2.4–2.7.

Probability: Definitions and Basic Concepts

Probability theory is concerned with predicting statistical outcomes. Simple examples of such outcomes include observing a head or tail when tossing a coin, or obtaining the numbers 1, 2, 3, 4, 5, or 6 when throwing a die. For a fairly-weighted coin, we would, of course, expect to see a head for 1∕2 of a large number of tosses; similarly, using a fairly-weighted die, we would expect to get a four for 1∕6 of all throws. We can then say that the probability of observing a head on one toss of a fairly-weighted coin is 1∕2 and that for obtaining a four on one throw of a fairly-weighted die is 1∕6. This heuristic notion of probability can be given mathematical formality via the following definition:

Here, sample space designates the available Ns occurrences while random event A denotes the subset of sample space given by Ne ≤ Ns.