We began by saying that uncertainty can be reasoned about exactly, but intuition cannot. We asked three questions: what exactly is the question, what is the clever way in, and what happens in the long run? Here is what the chapter’s twenty-five questions showed.
1. What exactly is the question?
The first four sections were about getting the question right. “Given that a numbered disc is taken” and “given that the disc numbered \(3\) is taken” gave different answers (Section 12.1). A \(90\%\) accurate test for a rare disease left a positive patient only \(9\%\) likely to be ill, because base rates matter (Section 12.2). “A random chord” had no expected length until we said how it was chosen (Section 12.3), and the broken stick (Section 12.21) made the same point. And “independent” turned out to have several meanings: events can be independent in pairs but not all together (Section 12.4), and uncorrelated variables need not be independent. In every case, the cure was to write down exactly which outcomes are possible and how they arise.
2. What is the clever way in?
Many questions had a long calculation and a short insight.
Symmetry. Two seats that the process cannot tell apart (Section 12.5); “more heads” and “more tails” as complementary mirror images (Section 12.6); every pair of players equally likely to contest any match (Section 12.7); HT and TH equally likely for any coin (Section 12.9); a radially symmetric point whose direction is uniform (Section 12.22).
Indicators and linearity. Counting pairs (Section 12.4), knots (Section 12.10) and matches (Section 12.7) as sums of indicators gave means and variances without the full distribution.
States and first-step analysis. Penney’s game (Section 12.8), the rabbit warren and Professor Ecks (Section 12.16), and the extinction of family names (Section 12.15) were all solved by conditioning on the next step.
Strategy. The worst shot in the truel should miss on purpose (Section 12.13); a randomised strategy cannot be exploited (Section 12.14); rejecting the first \(\frac1e\) of the candidates is optimal (Section 12.12).
3. What happens in the long run?
The last parts of the chapter followed randomness over time and across many trials. A branching process either dies out or explodes, and even an exploding one can die by chance (Section 12.15). A Markov chain settles into a stationary distribution and forgets its past (Section 12.16). Exponential waiting times have no memory at all (Section 12.18). Averages usually settle down, but not for the Cauchy distribution, which has no mean (Section 12.22). Sums of many independent effects become normal, by the central limit theorem (Section 12.24), and that normality hides Stirling’s formula for \(n!\) (Section 12.23). Maxima behave differently again: they grow like \(\ln n\) with a spread that does not grow at all (Section 12.25). And the constants \(e\), \(\pi\) and \(\gamma\) kept appearing, in cups and stamps, needles and loops.
4. A toolkit for probability
| Method | Where you used it |
|---|---|
| Write down the sample space | Conditioning (12.1); Bertrand (12.3) |
| Bayes’ theorem and odds | Base rates (12.2); Markov reversal (12.16) |
| Symmetry | 12.5, 12.6, 12.7, 12.22 |
| Indicators and linearity | 12.4, 12.7, 12.10 |
| First-step analysis | Penney (12.8); warren (12.16); branching (12.15) |
| Complement and bounds | Fair decisions (12.9) |
| Series for expectations | Stamps and \(e\) (12.11) |
| Parametrise and optimise | Secretary (12.12) |
| Game-theoretic reasoning | Truel (12.13); mixed strategies (12.14) |
| Counting paths to the end | Matchboxes (12.17) |
| Distribution function first | 12.18, 12.19, 12.25 |
| Probability as area or integral | Buffon (12.20); broken stick (12.21) |
| Normal approximation | Stirling (12.23); CLT (12.24) |
Where next
Chapter 13 turns from probability to statistics: estimating unknown quantities from data, starting with Section 13.1.
The counting problems of Section 9.8, Section 9.9 and Section 9.11 are the combinatorial side of this chapter.
Markov chains use the linear algebra of Chapter 7, and the fixed points of branching processes the dynamics of Section 8.7.
Joseph Blitzstein & Jessica Hwang, Introduction to Probability: a superb first course, free online, with video lectures.
Geoffrey Grimmett & Dominic Welsh, Probability: An Introduction: concise and rigorous, at first-year level.
Frederick Mosteller, Fifty Challenging Problems in Probability: classic puzzles with full solutions.
David Spiegelhalter, The Art of Statistics and The Art of Uncertainty: probability and risk in the real world.
Laplace called probability common sense reduced to calculation. This chapter has shown that the calculation is often needed precisely because common sense fails, and that the best calculations do more than produce numbers: they explain. A symmetry, an indicator, a well-chosen state or a limit theorem turns a puzzle into an insight, and those insights are what a university course in probability is built on.