2 Lecture 1, January 08, 2024

In this lecture, we went over

  1. Course syllabus and rules
  2. Chapter 1 – Basic definition of probability. We also saw the potential ambiguities when defining probabilities.

Definition 2.1 (Classical Definition of probability) The classical definition: The probability of some event is \[ \frac{\mathrm{number~of~ways~the~event~can~occur~}} {\mathrm{{the~total~number~of~possible~outcomes}}}, \] provided all outcomes are equally likely.

Definition 2.2 (Relative Frequency Definition of of probability) The relative frequency definition: The probability of an event is the (limiting) proportion (or fraction) of times the event occurs in a very long series of repetitions of an experiment.

Definition 2.3 (Subjective Definition of Probability) The subjective definition: The probability of an event is a measure of how sure the person making the statement is that the event will happen.

Problem: Each of the above definitions has pitfall:

  • Classical: We may not be able to know the total number of possible outcomes, or it may be uncountable
  • Relative frequency: We need “repetition”, which is often expensive and may not be possible.
  • Subjective: We want the probability to be consistent across different people, and and be rigorously defined.