Course contents document · University · generated 2026-07-15

Probability & Statistics

11 core topics + 22 prerequisite topics taught as needed · approximately 8 hours of instruction including spaced review

How the course runs

An adaptive diagnostic (up to 40 questions) places the student on the course's knowledge graph — topics already known are credited, and instruction begins exactly at the learning frontier. Every topic is taught with a worked-example lesson and auto-graded practice; a topic is mastered at 75%+ and then maintained through spaced reviews on an expanding schedule. A cumulative quiz follows every 6 lessons. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.

Core curriculum

Probability & Statistics · 11 topics

Mean, Median & Range [E] Center and spread in one pass.
Variance & Standard Deviation [M] Average squared distance from the mean — then unsquare.
Probability Basics [E] Favorable over total, when outcomes are equally likely.
Addition Rule & Complements [M] P(A or B) adds, then removes the double-counted overlap.
Conditional Probability [M] Probability after narrowing to a sub-population.
Independence & the Multiplication Rule [M] Independent events multiply.
Bayes' Theorem [H] Reverse a conditional: update a prior with evidence.
Expected Value [M] The probability-weighted average outcome.
The Binomial Distribution [H] Count the arrangements, multiply the per-path probability.
Normal Distributions & z-Scores [M] Standardize, then use the 68–95–99.7 rule.
Correlation & Regression Lines [M] The best-fit line predicts; its slope is the per-unit effect.

Prerequisite material — taught automatically when the diagnostic finds gaps

Arithmetic Foundations · 8 topics
Adding & Subtracting Whole Numbers Multi-digit addition and subtraction.
Multiplication Multiplying whole numbers.
Division Dividing whole numbers.
Order of Operations Parentheses first, then multiplication/division, then addition/subtraction.
Negative Numbers: Adding & Subtracting Working with numbers below zero on the number line.
Negative Numbers: Multiplying & Dividing Sign rules for products and quotients.
Exponents Repeated multiplication in shorthand.
Square Roots Undoing a square.
Fractions · 4 topics
Equivalent Fractions Different fractions can name the same amount.
Simplifying Fractions Reducing a fraction to lowest terms.
Multiplying Fractions Multiply straight across.
Dividing Fractions Multiply by the reciprocal.
Decimals, Percents & Ratios · 1 topics
Decimal Addition & Subtraction Line up the decimal points.
Expressions & Equations · 1 topics
Evaluating Expressions Substituting a value for a variable.
Linear Functions · 3 topics
The Coordinate Plane Locating points with (x, y) pairs.
Slope of a Line Rise over run between two points.
Slope-Intercept Form y = mx + b describes a whole line.
Math Olympiad: MOEMS & AMC 8 · 3 topics
Permutations Ordered arrangements.
Combinations Unordered selections: n choose k.
Counting & Probability Favorable outcomes over total outcomes.
Discrete Mathematics · 2 topics
The Product & Sum Rules Multiply independent stages; add exclusive alternatives.
Permutations & Combinations Ordered vs. unordered selections.

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