Probability¶

Open Frontier

What you will develop¶

Quantify uncertainty, conditioning, expectation, dependence, transformations, concentration and asymptotics.

Before you begin¶

Recorded prerequisites:

• Mathematical reasoning and proof
• Calculus + vector calculus
• Linear algebra

Study and practice¶

Progress by the work you can do, not by a fixed calendar.

Practice sequence 20¶

Spine: Blitzstein & Hwang, Introduction to Probability, 2e

Reading: Ch. 1, pp. 1-44: probability and counting

Know: Define sample spaces/events; count without over/undercounting; move between symmetry stories and formal probability.

Reconstruct: Derive combinations/permutations and at least one story proof.

Do: Estimate collision/failure probability in a large distributed-agent ID scheme analytically and by Monte Carlo.

Defend: When does 'equally likely' silently fail?

Gate: 8 counting/probability problems with no formula sheet; explain each sample space.

Source: source

Practice sequence 21¶

Spine: Blitzstein & Hwang, 2e

Reading: Ch. 2, pp. 45-102: conditional probability, Bayes, independence

Know: Update beliefs with evidence; use total probability/Bayes; distinguish independence from mutual exclusivity.

Reconstruct: Derive Bayes from conditional probability definition and total probability.

Do: Analyze a rare-failure diagnostic sensor; produce posterior failure probability under multiple base rates.

Defend: Why does a highly accurate test not imply a highly accurate positive prediction?

Gate: Closed-book diagnostic/base-rate oral defense.

Source: source

Practice sequence 22¶

Spine: Blitzstein & Hwang, 2e

Reading: Ch. 3-4, pp. 103-212: random variables, distributions, expectation, variance

Know: Model uncertainty with random variables; exploit linearity of expectation; reason with variance and indicators.

Reconstruct: Derive E[aX+b], Var(aX+b), and indicator-variable expectation method.

Do: Estimate expected downtime of a redundant robot fleet with shared and independent failure modes.

Defend: Why can expectation be useful even when the full distribution is hard?

Gate: solve one problem primarily with indicators and one with distribution conditioning.

Source: source

Practice sequence 23¶

Spine: Blitzstein & Hwang, 2e

Reading: Ch. 5-7, approx. pp. 213-366: continuous RVs, moments, joint distributions

Know: Work with densities, Normal/Exponential models, covariance/correlation, joint/marginal/conditional distributions.

Reconstruct: Derive covariance identity; derive convolution intuition for sums.

Do: Simulate correlated sensor errors; show when averaging sensors fails to reduce uncertainty as 1/sqrt(n).

Defend: What does zero correlation fail to tell you?

Gate: Unseen joint distribution: compute marginals/conditionals/covariance and interpret dependence.

Source: source

Practice sequence 24¶

Spine: Blitzstein & Hwang, 2e

Reading: Ch. 8-10, pp. 367-496: transformations; conditional expectation; inequalities & limit theorems

Know: Transform variables, condition on information, use bounds/LLN/CLT and know approximation regimes.

Reconstruct: Derive law of total expectation; prove Markov/Chebyshev inequalities; state LLN vs CLT distinctly.

Do: Build a Monte Carlo uncertainty estimator and empirically verify convergence/breakdown under heavy tails.

Defend: What does the CLT not guarantee?

Gate: Module gate: approximate a hard probability three ways-simulation, bound, asymptotic approximation-and reconcile differences.

Source: source

Exit gate¶

Closed-book: 120 min: counting, Bayes, expectation by indicators, covariance, conditional expectation, LLN/CLT/bounds.

Novel problem: Estimate risk for a redundant system with common-cause and independent failures.

Artifact: Monte Carlo simulation with convergence diagnostics and analytic comparison.

Defend: Explain sample space, conditioning information, independence assumptions and approximation regime.

Pass criterion: Pass if base rates/dependence are never silently omitted and simulation agrees with analytic results.

Transfer problems¶

Try these before consulting solutions or asking for the complete answer.

1. Counting: Compute collision probability for randomly assigned IDs and validate with simulation.

2. Bayes: Update a rare-event failure probability after a positive diagnostic with known sensitivity/specificity.

3. Independence: Construct variables that are pairwise independent but not jointly independent.

4. Indicators: Use indicator variables to derive expected number of failed components.

5. Variance: Compare uncertainty of average sensor reading under independent vs correlated noise.

6. Continuous: Derive and simulate waiting time behavior under an exponential assumption.

7. Joint: Given joint density/table, compute marginal/conditional distributions and covariance.

8. Conditional expectation: Compute E[X|Y] in a small model and interpret it as information-adjusted prediction.

9. Bounds: Use Markov/Chebyshev to bound risk where exact distribution is unavailable.

10. CLT: Simulate when CLT approximation becomes good/bad across light- and heavy-tailed examples.

Textbooks¶

See the five-book resource page.

Reading options¶

Assigned readings, where available, appear in the practice sequence above. These additional recommendations are not all required reading.

• Introduction to Probability

Continue¶

Connections above reflect recorded prerequisites, not a compulsory calendar.

Integrated practice | Choose a study route