Every quantitative claim in finance — a 5% VaR, a 60% chance a trade is profitable, a fair value for an option — is a probability statement, and a probability statement only means something once you have fixed what set of outcomes it is a statement about. This note builds that foundation: the sample space as the exhaustive, mutually-exclusive list of outcomes an experiment can produce, events as subsets of that outcomes may or may not satisfy, and the three Kolmogorov axioms that any assignment of numbers to events must satisfy to be called a probability law. Nothing here is optional machinery — every later result in this unit (a distribution, a random variable, a limit theorem) is a probability law on some sample space, and every property you will ever prove about one reduces, eventually, to these three axioms.
Learning objectives. By the end of this note you should be able to:
- construct an appropriate sample space for a given random experiment and identify the event algebra built from it (union, intersection, complement);
- state the three Kolmogorov axioms and verify whether a proposed probability assignment satisfies them;
- derive elementary consequences of the axioms (e.g. , finite additivity over more than two events) directly from the axioms;
- build a discrete probability law from an equally-likely-outcomes assumption and compute event probabilities from it.
In this lesson
- Sample spaces and events
- The probability axioms
- Constructing a discrete probability law
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