Every pricing model, every risk report, and every backtest is a claim about how likely something is — and that claim only means something once "likely" has been pinned down precisely. This topic builds the five tools every later result in Probability I leans on: the sample space and event vocabulary for describing an experiment, the Kolmogorov axioms that make "probability" a mathematically consistent object rather than a vague word, the counting techniques for computing probabilities when outcomes are equally likely, conditional probability and Bayes' rule for updating beliefs on new information, and independence for recognising when two pieces of randomness don't interact. Every subsequent topic in this unit — random variables, distributions, limit theorems — is built on top of these five ideas without restating them.
By the end of this topic you should be able to:
- write down the sample space and an event algebra for a given experiment, and check a candidate probability assignment against the Kolmogorov axioms;
- count outcomes with the multiplication principle, permutations, combinations, and multinomial coefficients, and use these counts to compute probabilities under equally-likely outcomes;
- apply the multiplication rule, the law of total probability, and Bayes' rule to compute and update conditional probabilities;
- test two events (or a collection of events) for independence, and distinguish pairwise from mutual independence;
- work a multi-stage probability problem end-to-end with a probability tree.
In this lesson
- Contents
- The big picture
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