Foundations — Probability — classical, empirical, axiomatic (Kolmogorov axioms)

Before you can read the parent note Probability, you must be able to read every squiggle it uses without flinching. This page builds each one from nothing. Read top to bottom — each symbol is earned by the one before it.

0. The starting picture: an experiment and its box

The whole subject lives inside one picture: draw a box. Every single thing that could happen is one dot inside that box. Nothing that happens can be outside the box — the box is, by definition, "all possibilities".

1. $S$ (also written $\Omega$) — the sample space

Picture: $S$ is the whole box from figure s01. Each dot inside it is one outcome.

• Coin: $S=\{H,T\}$ — two dots.
• One die: $S=\{1,2,3,4,5,6\}$ — six dots.

Why the topic needs it: every probability question secretly begins "out of what total set of possibilities?" — that total set is $S$.

2. Braces $\{\}$ and "element of" $\in$ — writing sets

Picture: $\in$ is the arrow "this dot lives inside that region". Look at figure s01 — the dot labelled $3$ satisfies $3\in S$.

Why the topic needs it: outcomes and events are sets, so we need the language of membership to say "the outcome that occurred is one of the favourable ones".

3. Outcome vs event, and the subset symbol $\subseteq$

Why the topic needs it: the thing we assign a probability to is an event (a region), not usually a single outcome. "Rolling even" is a region of three dots.

4. Two special regions: $\varnothing$ and $S$ itself

Picture: $\varnothing$ is an empty circle; $S$ is the entire box shaded. Every other event sits somewhere between these two extremes — and that "between-ness" is exactly why probability will run from $0$ (for $\varnothing$) to $1$ (for $S$).

5. The three combining operations: $\cap$, $\cup$, $A^c$

This is the heart of "computing with events". Human logic words — and, or, not — become three picture-operations.

Why the topic needs it: the addition rule $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ and the complement rule $P(A^c)=1-P(A)$ are literally statements about these three pictures.

6. Mutually exclusive (disjoint) — $A\cap B=\varnothing$

Picture: two separate blobs in the box with a gap between them (no overlap). "Rolling a $2$" and "rolling a $5$" can't both happen on one roll → disjoint.

Why the topic needs it: Kolmogorov's third axiom (additivity) only applies to disjoint events — for them you simply add the probabilities, no overlap to worry about. Miss this word and you'll wrongly add overlapping events.

7. The measuring symbol: $P(A)$

Picture: think of shading. If $A$ covers a big share of the box, $P(A)$ is near $1$. If tiny, near $0$. $P(\varnothing )=0$ (no box), $P(S)=1$ (whole box).

Why the topic needs it: this symbol is the topic. Everything else exists so that $P(A)$ has a clear, computable meaning.

8. Counting symbols: $n$, $m$, $f$, $N$ and the fraction bar

Picture: $\frac{m}{n}$ = (shaded dots) ÷ (all dots). $\frac{f}{N}$ = (successful trials) ÷ (all trials). Both are "part over whole", which is why both live between $0$ and $1$.

9. The infinity tools: $\lim$, $\sum$, $\bigcup$

These appear in the empirical definition and the additivity axiom. Meet them before you meet the formulas.

Why the topic needs it: without $\lim$, "empirical probability" would be a different (noisy) number every time you tried. The limit gives it a single, stable meaning.

10. Number-set symbols: $\mathbb{R}$, and $0\le P(A)\le 1$

Picture: a number line from $0$ to $1$; every probability is a point on that short segment. $0$ = impossible end, $1$ = certain end.

Why the topic needs it: the axioms are stated as "a function $P$ from events to $\mathbb{R}$" — you must know $\mathbb{R}$ means "the number line" to parse that sentence.

How the foundations feed the topic

Read it as: an experiment gives a sample space of dots; groups of dots are events; combining and measuring events, plus counting or measuring trials, all pour into the three axioms — from which everything else is proven.

Equipment checklist

Test yourself — cover the right side and answer aloud. If any is shaky, reread that section.

Connections

• Set Theory — Union, Intersection, Complement — the picture-algebra of $\cap ,\cup ,A^c$ used everywhere above.
• Law of Large Numbers — guarantees the limit in section 9 actually settles.
• Conditional Probability & Bayes' Theorem — the next layer, once $P(A)$ is solid.
• Independence of Events — built from $\cap$ and $P$.
• Random Variables & Probability Distributions — attach numbers to the dots of $S$.
• Permutations & Combinations — the machinery for counting $m$ and $n$ in the classical formula.