Start learning probability with a coin

Probability
rules them all.

Why don't ten tosses guarantee five heads? How can frequent wins still leave you losing money? Start with small questions, learn to calculate chances, and consider what the outcomes cost.

Read Chapter 1 Try a random walk

Basic arithmetic is enough to begin.
Inspired by Taleb's thinking on risk: examples first, formulas explained.

List the outcomes·State the conditions·Ask what you can afford to loseP(A)

I The main course

Probability Made Easy
with Taleb

Sixteen chapters, three exercises each, and answers at the end. Read from the start or choose a question that interests you. The experiments below accompany the book.

Part I / Everyday questions01 What probability can tell us 02 List the possible outcomes 03 Why ten tosses need not give five heads

Part II / Basic methods04 When to add and when to multiply 05 How to calculate “at least once”06 New information, new probabilities 07 Independent or mutually exclusive?08 A positive test: how likely is illness?

Part III / Common mistakes09 Why shared birthdays are not so rare 10 Three doors: should you switch?11 An average gain is not a promised gain 12 Five heads in a row: what next?

Part IV / Making judgments13 Fair rounds can still leave you broke 14 Probability 1 can allow exceptions 15 Rare, large losses: understanding fat tails 16 Before deciding, ask what you can afford to lose

II A classic problem in Feller

Every step is fair.
Why can you still go broke?

Start with 10 chips. Win or lose one per step, stopping at 20 or 0. The default win probability is 50%; move the slider to change it and compare outcomes.

Random walk / ruin and targetStart: 10 chips

Current path

This gameStart: 10Target: 20Ruin: 0

Win probability per step50%

III An easily confused idea

Probability 1:
how can exceptions remain?

Choose a real number uniformly from 0 to 1. Selecting exactly 0.5, specified in advance, has probability 0. Yet 0.5 is still an available outcome. Probability zero and an empty event are different things.

Almost surely

If event A has probability 1, its failure has probability 0. But there may still be outcomes where A fails. “Not selecting 0.5” has probability 1 while excluding the possible outcome 0.5.

Every face of a fair die has positive probability, so a probability-1 event must include them all. In a continuous model, a point has no length and can have probability 0.

Diagram: a specified point has probability zero under uniform sampling from the real interval from 0 to 1; the whole interval has probability one. — This diagram assumes uniform sampling from the real interval from 0 to 1. “Ignored” means assigned probability zero, not removed from the possible outcomes.

Try another example

An exception can exist with probability zero.

P(A) = 1

Uniform choice from 0 to 1

A number is chosen, but a specified point has probability 0.

Here probability follows interval length. A point has no length, so choosing exactly the prespecified number 0.5 has probability 0. But it remains available, and every draw selects a particular number.

P(X = 0.5) = 0　·　P(X ∈ [0,1]) = 1

Zero means exactly zero here. Why can every point have probability 0 while the interval has probability 1? Chapter 14 explains why not all infinite collections can be summed like finite ones.

Read Chapter 14: probability 1 can allow exceptions ↗

A A theological extension

Does this mean the elect could finally be lost?

The analogy does not transfer directly

“Probability 1 can allow exceptions” is a claim within a probability model. Classical Reformed confessions instead say that true believers may fall temporarily but will not finally be lost. The shared language of certainty does not make these the same claim.

Distinguish whether someone truly is elect from whether we can know that accurately. Chapter 14 contains the full discussion, a comparison of terms, and the confessional sources.

Read the discussion ↗Westminster Confession, Chapter 17 ↗Canons of Dort, fifth head ↗

IV Counterintuitive experiments

Try it first.
Then see why.

Play the three-door game repeatedly, or change the three percentages in the testing example. Observe how the results change, then turn to the book for the calculation.

LAB 02

Three doors

Choose, then decide whether to switch

A prize is placed randomly behind one door; the other two are empty. Choose a door. The informed host always opens an unchosen empty door and offers you the chance to switch.

Choose a door first.

The key idea

The host knows where the prize is and must avoid revealing it.

Simulation results will appear here.

LAB 03

After a positive test

Count the people

Suppose 10,000 people are tested. Move the sliders to see how many positive results are genuine cases and how many are false alarms.

Prevalence1% Positive when ill99% Positive when not ill5%

Among positive results, the proportion genuinely ill16.7%

The key idea

The fraction of cases detected is different from the fraction of positives who are genuine cases.

These are invented teaching figures, not data about a real disease or test.Base rates: Chapter 8 · Three doors: Chapter 10

V / A question about risk

Knowing the chance is only part of the decision.
If the outcome is bad, can you afford it?

Read with this question in mind

VI Further reading

After this book,
what comes next?

Use this introduction for the basics, then explore Feller's probability textbook or further study in statistics, risk, and related fields.

Main textbook / 16 chapters · 48 exercises

Probability Made Easy with Taleb

From coins and dice
to better judgments.

Sixteen chapters, 48 exercises, answers, and reading suggestions. Look up unfamiliar terms in the glossary at the end.

Open the book

16chapters48exercises

Screenshot of an X exchange: after recommendations of Feller and Ian Hacking, Nassim Nicholas Taleb replies “No Hacking.” — **What does “No Hacking.” mean?**In the context of the reading list, the natural interpretation is that he objects to including Hacking. The reply gives no reason. The appendix discusses what can and cannot be inferred.Sources and explanation in the book ↗

05 Choose by interest

Taleb's library
A provisional reading list

Compiled from a shelf-identification list supplied to this project, not yet verified book by book. Each entry includes an introduction, prerequisites, and this site's rating.

About the listTaleb wrote Dynamic Hedging and the fat-tails monograph listed here. Owning a book does not establish a recommendation, and the ratings below are not his ratings.Taleb's CV ↗Fat-tails research ↗

Analyze data over time

Time Series Analysis

Study patterns in monthly or yearly observations. Prior statistics and calculus are recommended.

Understand complex change

Nonlinear Dynamics and Chaos

Explore complex behavior through oscillation, growth, and other concrete examples. Some calculus is needed.

Study statistical inference

Statistical Inference

Learn how samples can inform estimates of populations and tests of hypotheses. Requires calculus and probability.

Study financial markets

The Econometrics of Financial Markets

Learn to test market theories with financial data. Best with a background in statistics and economics.

Understand rare, large losses

The Statistical Consequences of Fat Tails

Study how extreme outcomes affect averages and statistical judgments. An advanced read requiring mathematical preparation.

Ratings are out of 5, based on relevance to this site's subjects: 5 means a priority recommendation, 4 recommended reading, 3 broader exploration, and 2 a specialist interest. A high rating does not mean an easy book: check the prerequisites. “Derivatives” has not been identified precisely and is rated provisionally by subject only.

Selected titles · filter by subject

Book index

Search by title or author

Titles and authors follow the supplied list; editions still need checking. “Derivatives” does not yet identify a specific book.

Understand one problem properly

List the possible outcomes,
then consider what each would mean.

Back to the coin experiment