--- name: math-thumb-probability description: "Apply Chapter 12 (Probability: Fast Reasoning Under Uncertainty) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to choose among an exact complement, approximation, and guaranteed probability bound." --- # Probability: Fast Reasoning Under Uncertainty Use this chapter to help the reader make a checked mathematical decision. All 30 numbered rules are available in [the chapter source](references/chapter.md). [The workbook](references/notebook.md) contains a lab, exercises, solutions, and the full rule checklist. [The local rule index](references/rules.json) supplies discovery metadata. ## Start from the reader's task Infer solve, learn, or audit mode from the request. In solve mode, use their supplied numbers and target; in learn mode, use the workbook or their chosen rule; in audit mode, inspect their actual calculation before replacing it. Gather only missing information that changes the choice: Event and threshold; distribution information; number of opportunities; dependence; moments; whether a guarantee or approximation is needed. If the question falls outside this chapter, say which mathematical operation is missing and suggest a relevant chapter. If the whole-book skill is available, it can carry the task onward, but this chapter works independently. ## Select and apply a rule - **Need a guaranteed bound with little distributional information?** Start with Markov, Chebyshev, Cantelli, or the union bound according to the available moments and tail direction. - **Know event overlaps?** Add the Bonferroni pairwise correction and decide whether the resulting bracket is narrow enough. - **Combining uncertainty?** Check covariance before using root-sum-of-squares, and condition explicitly before splitting total variance. - **Facing repeated rare opportunities?** Use the exact complement first, then recognize Poisson, birthday, coupon, or exponential scales only in their regimes. - **Updating evidence?** Convert the base rate to prior odds and multiply only conditionally independent likelihood ratios. - **Replacing a distribution?** Check support, expected successes and failures, skew, moments, and tail location before selecting a normal, Poisson, or exponential approximation. - **Transforming an estimate?** Use Jensen for bias direction, the lognormal formula for an exact normal-log model, or the delta method for local uncertainty propagation. - **Need sharper tail control?** Ask what extra fact is defensible: direction, a hard bound, total variance, or sub-Gaussian decay. - **Running a simulation?** Report root-$N$ Monte Carlo error, then audit correlation, bias, weight concentration, and proposal support. Read the selected complete profile, including its equation, “How to read it,” and “How to use it.” The compact graph assumptions are search cues, not a substitute for the profile. Preserve the numbered citation and role. **Independent**, **Workflow**, and **Specialized** describe the relationship to a calculation; exactness, approximation, bound, diagnostic, and heuristic describe a different dimension. Use verified inputs, show the substitution and units, and interpret the result in the reader's decision. Verify by an appropriate bound, alternative computation, limiting case, residual with conditioning, or sensitivity check. If a required condition fails, reject that use and give the specific missing information or alternative method; do not calculate a plausible-looking answer from an invalid formula. **Essential boundary:** Do not silently multiply probabilities or likelihood ratios when the required independence is unverified. Distinguish bounds from approximate event rates. For a sufficient independent result, stop with the decision it supports. For a workflow or specialized rule, name the downstream calculation still needed. A numerical demonstration is evidence for that instance, not a universal proof. ## Teach and check understanding Use the [workbook](references/notebook.md) for guided practice. Start with 12.1.3, 12.1.8, 12.2.1 when the reader wants a starting exercise. Ask for an attempt, offer a relevant hint, and reveal the answer when requested or when teaching requires it. Do not force a quiz when the reader asked for a worked solution. Check whether the reader can explain the controlling quantity, apply the rule to a changed input, identify an invalid use, and distinguish a final answer from a preparatory step. Track only demonstrated work. Give a short prerequisite explanation when needed; avoid requiring completion of earlier chapters. ## Return a usable result Include the chosen rule numbers, assumptions that matter, calculation, verification, and next action. For ongoing work, offer this compact record: question; inputs and units; rules; claim type; book role; assumption status; result and error; check; decision; unresolved next step. Write a progress file only when asked or within an already authorized notebook-editing task. The source chapter is a fixed book snapshot. Preserve its mathematical qualifications and historical evidence gaps. Use outside material only when the reader's task needs it, verify material facts appropriately, and identify that material separately from the book. ## Illustrated exploration Open [the browser reader](assets/reader.html) or [the saved illustrated notebook](assets/notebook.ipynb). [Equation cards](references/equations.json) record the book rule, formula, fixed inputs, supported choices, assumptions and executed default results. - **C12-D01: Accumulate a rare risk across opportunities**: rule 12.1.8. - **C12-D02: Make the base rate visible in an update**: rule 12.2.1. - **C12-D03: Run a finite Monte Carlo estimate**: rule 12.3.7. - **C12-D04: See how soon random labels collide**: rule 12.1.9. - **C12-D05: See why the last coupons take longest**: rule 12.1.10. Use a saved illustration only when its conditions fit. Browser controls select finite precomputed choices; they do not calculate arbitrary reader inputs. For different inputs, make a checked calculation using the selected rule. Explain what changes, and never claim the notebook ran or the browser was viewed unless it did. Offer prediction questions for learning; answer direct requests without a mandatory quiz.