Suppose you roll two dice. Of course, you can’t say exactly what you will get. But there are some things you can say. You can say, for example, that it is less likely for the dice to land “snake eyes” (two ones) than to not land snake eyes. You can say that getting a sum of two is as likely as getting a sum of twelve. And you can say that there is no way you will get a sum of one. Probability is a mathematical discipline which formalizes and systematizes these intuitive judgments. And probability has been used to great effect in a remarkably wide variety of intellectual endeavors.
Few mathematical concepts are as useful as the concept of probability - it is ubiquitous in all the physical and social sciences, as well as being the central subject of a robust field of mathematics. Yet despite its indispensability, there is still a deep mystery about what probabilities actually are, and about what we are doing when we use probabilities. In this class, we will examine this mystery by first getting clear on the mathematical theory of probability, and then considering different philosophical interpretations of the mathematical theory.
As the philosopher Bishop Butler famously said, “probability is the very guide of life.” Still, however, it is unclear how to understand probability statements. What does it mean to say that getting snake eyes is less likely than not getting snake eyes? Maybe it means that if you rolled the two dice many times, then in the long run you will get snake eyes a lot less often than not getting snake eyes. This is a “frequentist” interpretation of probability. But such an approach to probability runs into problems. What about cases where we make probability judgments about events that cannot be repeated? For example, suppose we say “it is likely that it will rain tomorrow” or “chances are the extinction of the dinosaurs were caused by a meteor.” Another way of thinking about probability – the “epistemic” interpretation – says that probability statements are about what is rational to believe, given a body of evidence. Epistemic probabilities might be able to explain these “one-off” probability judgments – but they run into problems of their own.
In this class, we will begin by getting clear on the probability calculus by introducing the standard axioms and deriving important results, such as Bayes’ rule. We will talk about how the probability calculus applies to everyday life, and explore ways in which people are very bad at probabilistic reasoning (such as the base rate fallacy and the Monty Hall problem).
By the end of this course, students will:
Prerequisites: Students need to have some familiarity with high school level algebra.