Prerequisites & Notation

Before You Begin

Chapter 1 is the starting point for the entire book — it has no prerequisites within the library. The material below should feel familiar from undergraduate study. If any item feels shaky, spend thirty minutes reviewing it before proceeding: the rest of the book builds on these foundations.

Set notation: $\in$ , $\subseteq$ , $\cup$ , $\cap$ , $A^c$ , $\emptyset$
Self-check: Can you write down De Morgan's law from memory and check it with a small example?
Elementary logic: and, or, not, implication
Self-check: Can you translate 'not (A and B)' into set notation?
Sums of geometric and telescoping series
Self-check: Do you know $\sum_{k=0}^{\infty} r^k = 1/(1-r)$ for $|r| < 1$ ?
Counting: factorials, binomial coefficients
Self-check: Can you compute $\binom{52}{5}$ without a calculator (leaving it as a formula)?
Basic proof technique: induction
Self-check: Can you prove the binomial theorem by induction?

Notation for This Chapter

Symbols introduced in this chapter. The notation follows Kolmogorov's original conventions, which are essentially universal in modern probability.

Symbol	Meaning	Introduced
$\Omega$	Sample space — the set of all possible outcomes of an experiment	s01
$\omega$	A single outcome (sample point), $\omega \in \Omega$	s01
$A, B, C$	Events — subsets of $\Omega$	s01
$\mathcal{F}$	Sigma-algebra — the collection of measurable events	s02
$2^\Omega$	Power set of $\Omega$ — all subsets of $\Omega$	s02
$\mathcal{B}(\mathbb{R})$	Borel sigma-algebra on $\mathbb{R}$	s02
$(\Omega, \mathcal{F}, \mathbb{P})$	Probability space — the central mathematical object of this chapter	s03
$\mathbb{P}(A)$	Probability of event $A$	s03
$n!$	Factorial: $n(n-1)\cdots 2\cdot 1$ , with $0! = 1$	s04
$\binom{n}{k}$	Binomial coefficient: $n!/(k!(n-k)!)$	s04
$\binom{n}{k_1,\ldots,k_r}$	Multinomial coefficient: $n!/(k_1!\cdots k_r!)$	s04

Back to chapter overview Sample Spaces and Events