Probability Studio

Intermediate → Olympiad

Probability isn't a list of formulas. It is choosing the right picture of the randomness in front of you. Each chapter builds one idea from its definition, puts it to work on problems, and then stretches it on a story.

1
Probability as area
Outcomes, events, and why a probability is a share of one square.
Representation: The unit square
Open →
2
Counting outcomes
Before any probability: what exactly are the outcomes?
Representation: Tables, trees and lists
Open →
3
Independence and multiplication
Why “and” multiplies, and when it doesn't.
Representation: Area times area
Open →
4
Complement and “at least one”
When counting explodes, count what didn't happen.
Representation: The rest of the square
Open →
5
Exactly k
One string's chance, times the number of ways to arrange it.
Representation: Success–failure strings
Open →
6
Conditional probability
Keep only the outcomes that fit what you saw, and count again.
Representation: Zoom into what you know
Open →
7
Bayes and total probability
From causes to evidence and back: what a positive test really tells you.
Representation: Split by the cause
Open →
8
Symmetry
Sometimes the answer needs no calculation at all.
Representation: Random orderings
Open →
9
States
The present holds everything that matters: walks, races and urns.
Representation: State diagrams
Open →
10
Expectation and linearity
Random variables, their average and their spread, and why expected totals add up even when the parts are dependent.
Representation: Indicators
Open →
11
Conditional expectation
Expected waiting times, first successes and the coupon collector.
Representation: State diagrams with values
Open →
12
Chance and data
From probability to samples: why estimates settle, and why the error shrinks like 1/√n.
Representation: Repeated experiments
Open →

Final round

Ten problems from the whole studio. Nothing tells you which picture to use.