The four preceding lessons built the toolkit — axioms, counting, conditioning, independence — as separate pieces. This capstone runs them together on multi-stage problems, the kind an interviewer or an exam actually asks: a sequential model that needs a probability tree to organise, the Monty Hall problem (the single most famous probability paradox, and a genuine test of whether Bayes' rule has actually been understood or just memorised), and a full Bayesian updating chain. There is no new theory here — the payoff is fluency: recognising which stage of the toolkit a given clause of a word problem calls for, and building the tree that makes the calculation mechanical rather than a fresh derivation each time.
Learning objectives. By the end of this note you should be able to:
- build a probability tree for a multi-stage random experiment and read off event probabilities by summing over leaves;
- solve the Monty Hall problem for the "always switch" and "always stick" strategies, and explain precisely where the standard wrong answer (treating the two remaining doors as equally likely) goes wrong;
- carry out a multi-step Bayesian update and interpret the posterior at each stage as the next stage's prior.
In this lesson
- The probability-tree method
- The Monty Hall problem
- A full Bayesian updating chain
Before you start
- Sample Spaces, Events, and the Probability Axioms
- Counting and Combinatorics
- Conditional Probability, Total Probability, and Bayes' Rule
- Independence
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