Lecture 1: Basic Probability

The instructor begins the course by addressing the common misconception that finance is easy, emphasizing the importance of understanding uncertainty in financial markets. They introduce the course's two-part structure, focusing first on probability and stochastics, then on financial commodity pricing. The necessity of grasping the language of uncertainty through probability theory is highlighted, as is the difficulty of defining randomness and the historical connection between probability theory and gambling, exemplified by the collaboration between Blaise Pascal and Pierre de Fermat.

This paragraph delves into the concept of randomness, defining it as the lack of prior knowledge of outcomes from certain activities. The instructor uses the example of dice throws to illustrate randomness and discusses the development of probability theory from gambling problems. The notion of a random experiment is introduced, with examples such as stock prices and car accidents, to emphasize the unpredictability of outcomes. The instructor also touches on Bayesian ideas and the importance of defining a sample space for probability discussions.

The paragraph discusses the basic assumptions of probability, such as knowing all possible outcomes of a random experiment, which could be finite or infinite. It contrasts the classical definition of probability, which is based on equally likely outcomes, with the need for an axiomatic approach to handle more complex scenarios, including infinite sample spaces. The instructor introduces the concept of a sample space and the limitations of the classical definition, pointing out its circular logic and inability to address certain cases.

The instructor critiques the classical definition of probability for its circular reasoning and its limitations in handling infinite sample spaces. They introduce the axiomatic formalism of probability developed by Kolmogorov, emphasizing the need to move beyond basic intuitions to a more rigorous mathematical framework. The paragraph sets the stage for a deeper exploration of probability in the context of both finite and infinite sample spaces, hinting at the paradoxes that can arise without a formal approach.

This paragraph presents Kolmogorov's axioms as the foundation for a formal understanding of probability. The instructor outlines the conditions that define a probability space, including the sample space, the sigma-algebra of events, and the probability measure itself. They explain the axioms that any probability measure must satisfy, such as the assignment of values between 0 and 1, the inclusion of the empty set and the entire sample space, and the rules governing the probability of unions of events.

The instructor discusses the practicality of Kolmogorov's axioms in both finite and infinite sample spaces. They highlight the importance of the sigma-algebra in defining events and the limitations of the classical definition when dealing with infinite outcomes. The paragraph also foreshadows the discussion of the Bertrand paradox and other historical problems in the next class, emphasizing the utility of the axiomatic approach in resolving such paradoxes and in constructing probability spaces.

In conclusion, the instructor summarizes the importance of the axiomatic approach to probability and its applicability to various sample spaces. They provide a preview of the next lecture, which will include problem-solving related to the Bertrand paradox, Buffon's needle problem, and the construction of probability spaces.