Add law of conditional probability, Bayes' theorem, LOTUS, and law of iterated expectations to probability chapter

[Copilot]

• Add law of conditional probability (definition + theorem + proof + example)
• Add Bayes' theorem (theorem + proof + example)
• Add law of total probability (theorem + proof)
• Add LOTUS (theorem + proof + notes + example)
• Add law of iterated expectation (theorem + proof + example)
• Add proofs of all theorems
• Address first round of code review feedback
• Generalize @def-probability property 3 to countable additivity
• Generalize @thm-total-prob to countable partition
• Add continuous case proof to @thm-lie
• Explain how countable additivity is stronger than finite additivity (notes div after property 3)
• Improve #def-cond-expectation: add conditional pmf/pdf definitions (Hogg-TZ style) for both discrete and continuous cases
• Add Wikipedia citation for the additive-but-not-σ-additive counterexample

[d-morrison]

@copilot include proofs of those theorems

[Copilot]

@copilot include proofs of those theorems

Done in 5702435. All five theorems now have proofs:

• #thm-law-conditional-prob — rearrangement of the definition of conditional probability
• #thm-total-prob — follows from mutual exclusivity of $A \cap B_i$ and the multiplication rule
• #thm-bayes — apply the definition of conditional probability to $\Pr(A \cap B)$ from both sides
• #thm-lotus (new) — let $Y = g(X)$ and expand $\E{Y}$ via @def-expectation, then rearrange the double sum over $x$ and $y$ values
• #thm-lie — expand $\E{\E{Y \mid X}}$ via @def-expectation and apply the law of total probability

[Copilot]

Pull request overview

This PR expands the Probability chapter with key foundational results (conditional probability, total probability, Bayes’ theorem, LOTUS, and the law of iterated expectations), including definitions, proofs, and illustrative examples to support later statistical modeling content.

Changes:

• Added conditional probability definition, the law of conditional probability, and an applied example.
• Added the law of total probability and Bayes’ theorem with proofs and a medical-test PPV example.
• Added LOTUS and the law of iterated expectations with proofs, notes, and examples.

[d-morrison]

@copilot apply changes based on the comments in this thread

[Copilot]

@copilot apply changes based on the comments in this thread

Applied all four review suggestions in 08385ab:

• probability.qmd:195 — Changed "By property 3" → "By repeated application of property 3 of @def-probability" in the @thm-total-prob proof.
• probability.qmd:433-438 — Scoped the LOTUS theorem (#thm-lotus) to the discrete case only; the continuous analogue is now presented as an unproved remark inside the notes block after the proof.
• probability.qmd:498-503 — Added #def-cond-expectation defining $\E{Y \mid X = x}$ (with discrete and continuous forms) and the conditional expectation function $\E{Y \mid X}$, placed immediately before #thm-lie.
• probability.qmd:510-513 / 517-520 — Narrowed the #thm-lie statement to "discrete random variables with finite ranges"; updated the proof intro to explicitly invoke @thm-total-prob for the finite partition ${X = x : x \in \mathcal{R}(X)}$ by repeated application of property 3.

[d-morrison]

@copilot see my comments above

[Copilot]

@copilot apply changes based on the comments in this thread

All comments from that review thread have been addressed in 905f483.

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