Track 1
Probability is the mathematics of uncertain worlds. The goal is not to memorize formulas — it is to model uncertainty, condition on useful information, exploit symmetry, and prove that randomness behaves predictably. Eleven modules, each a set of concepts, mental moves, and the traps that catch adults.
Four sittings a week, 20–30 minutes each. One goal per week, one boss mini-problem to close it.
| Week | Goal | Boss mini-problem |
|---|---|---|
| 1 — Sample space & counting Stop counting the wrong universe. Ordered vs unordered, complements, combinations. |
Choose the universe deliberately before computing anything. | Birthday collision: exact vs union-bound approximation. |
| 2 — Conditioning & Bayes Information changes the universe. Filtering, observation traps, natural frequencies. |
Condition on what was observed, not what you wish were observed. | Two Monty Halls with different host behavior. |
| 3 — Independence & random variables Independence is information transfer, not surface unrelatedness. |
Choose a distribution from its generative story. | Pairwise independent but not mutually independent, by construction. |
| 4 — Expectation & indicators Linearity beats enumeration. Fixed points, empty boxes, collisions, inversions. |
Replace one impossible global count with many easy local ones. | Expected inversions in a random permutation. |
Every concept cluster maps to a frame you already reason in:
| Arc | Concepts | The question it answers |
|---|---|---|
| Risk desk | expectation, variance, concentration | “What loss should we expect, and how bad can it get?” |
| Hiring funnel | conditional probability, Bayes | “What does a signal really tell us?” |
| System reliability | independence, union bound | “How often does the system fail?” |
| Game table | symmetry, conditioning, martingales | “What can a fair game guarantee?” |
| A/B testing | distributions, variance, inference | “How much evidence is enough?” |
| Random construction | probabilistic method | “Can randomness prove a deterministic object exists?” |