Mathematics · intro

Probability Fundamentals

Sample spaces, conditional probability, independence, and Bayes' rule for updating beliefs.

mathematicsprobability

Mental model

Probability is a bookkeeping system for uncertainty. Conditional probability restricts the sample space; Bayes flips cause and effect — from 'how likely is the evidence given the hypothesis' to 'how likely is the hypothesis given the evidence'.

How to study Probability Fundamentals

Begin by restating the mental model in your own words, then connect it to a concrete system you have built or operated. Name the mechanism, the constraint it addresses, and the trade-off it introduces. Use Khan Academy — Probability, MIT 6.041SC — Probabilistic Systems Analysis (course), Seeing Theory — Basic Probability to check details, but close the source before writing your explanation. Retrieval is the learning step; rereading is only preparation.

Next, compare Probability Fundamentals with Random Variables & Distributions, Classical Distributions, Hypothesis Testing, ML Math Foundations. Ask what changes in correctness, latency, resource use, operability, and failure recovery. Complete Apply Bayes' rule, Independence and inclusion-exclusion and preserve the command, input, output, and one failed attempt as evidence. Finish by explaining the idea without jargon to someone who has not studied the track.

Proof of understanding

Explain the mechanism from first principles and identify the state it reads or changes.
Give one situation where the concept is the right choice and one where it is not.
Predict a realistic failure mode before running the drill, then compare the prediction with evidence.
Connect the result to a roadmap or build artifact instead of treating the concept as isolated trivia.

Where it matters

A/B test analysis, spam filtering, medical diagnosis, model calibration.

Common mistakes

Treating independent events as mutually exclusive
Confusing P(A|B) with P(B|A) (the prosecutor's fallacy)
Adding probabilities of overlapping events without inclusion-exclusion

Learn from primary sources

Khan Academy — Probability (course)
MIT 6.041SC — Probabilistic Systems Analysis (course) (course)
Seeing Theory — Basic Probability (article)

Practice and explain it back

Apply Bayes' rule

A test has 99% sensitivity and 95% specificity. Disease prevalence is 1%. A patient tests positive. Using Bayes' rule, compute P(disease | positive) to 2 decimal places.

Expected evidence: ≈ 0.17 (17%). Most positives are false alarms when the base rate is low.

Open the interactive drill →

Independence and inclusion-exclusion

P(A)=0.4, P(B)=0.5, P(A∩B)=0.2. Are A and B independent? Compute P(A∪B). If they were mutually exclusive, what would P(A∪B) be instead?

Expected evidence: P(A)P(B)=0.2=P(A∩B) → independent. P(A∪B)=0.4+0.5−0.2=0.7. Mutually exclusive would give 0.9 — conflating the two is a common bug.

Open the interactive drill →

Review prompts

State Bayes' rule and when you need it.
How is independence different from mutual exclusivity?

Build evidence

Simulate coin flips, dice, and random walks

Code simulators (no stats libraries): Bernoulli trials, dice sums, and a 1D random walk. Plot or print distributions and compare sample mean to theory.

10k coin flips: sample proportion within 0.02 of 0.5
Dice sum histogram matches triangular shape qualitatively
Random walk: report mean displacement and max |position| over T steps

Weekly math writeup (public note)

Explain one concept you drilled this week in your own words — derive or prove one step, include one mistake you caught yourself making.