SYSTEMATIC MATHEMATICS

Probability & Statistics

This course contains 58 visible knowledge chapters across nine content-sized units of lengths 5, 5, 6, 5, 5, 5, 8, 11, and 8. Learn probability and statistics from questions a beginner can see: what can happen, how likely is each result, what changes after new information, what does an average summarize, and how uncertain is a conclusion drawn from a sample? The course starts with coins, dice, cards, finite tables, and ordinary fractions. Every new symbol is read aloud and tied to one concrete quantity before a formula is used. It then builds probability tables, conditioning trees, random variables, order statistics, discrete and continuous distributions, convolution, joint models, repeated-sample behavior, Markov and Chebyshev bounds before the law of large numbers, standard errors, t inference, paired and independent comparisons, bootstrap and permutation reasoning, likelihood, chi-square tables, ANOVA, Bayesian updating and posterior prediction, intervals, tests, multiplicity, multiple and logistic regression, residual diagnostics, mean and individual prediction intervals, held-out validation, confounding, missingness, randomized comparisons, and the boundary between association and causation. The final ninth unit reconstructs a small study from its raw data rows and keeps arithmetic, assumptions, uncertainty, and unsupported claims separate. Time series, survival analysis, complex surveys, spatial statistics, high-dimensional learning, Bayesian computation, and advanced causal identification remain explicit later-course boundaries rather than hidden omissions.

Before this course: Completed Algebra & Functions, including fractions, ratios, equations, functions, graphs, and elementary sequences. Finite-set, counting, summation, exponential, area, and integral notation are introduced or reviewed before use. No calculus, proof course, programming, spreadsheet, statistics package, normal-table memorization, or prior probability course is assumed at the opening.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

A probability space separates outcomes, events, and assigned probabilities

Objective: Why is this not the ordinary fair-die answer by default?

One die roll produces exactly one outcome such as 4. An event is a question represented by a group of outcomes, such as “the result is even,” represented by {2,4,6}. A probability model assigns one number from zero to one to each possible outcome; for discrete outcomes that number is formally called probability mass, but it means probability, not physical weight.

What must be established: For every event A, 0≤P(A)≤1, P(∅)=0, P(Ω)=1, and P(Aᶜ)=1−P(A). First use the definition “For a finite model, Ω is the complete outcome list and A is any selected group inside it. The notation p(ω) is the probability of one outcome ω. All individual probabilities must be nonnegative and add to one. The event probability P(A) is found by adding p(ω) only for outcomes belonging to A.” The first nearby case where the conclusion can fail is: Listing only “success” and “failure” is invalid when the experiment’s complete outcomes or their probabilities are not defined.

Read the notation before calculating: Read Ω as “the list of all possible results.” A is one chosen group of results. The symbol ω means one particular result inside Ω. P(A) means “the probability that the result belongs to A.” The notation p(ω) means “the probability assigned to that one result.” Σ means “add the following quantity once for every listed result”; it is an instruction to add, not a new unknown.

For every event A, 0≤P(A)≤1, P(∅)=0, P(Ω)=1, and P(Aᶜ)=1−P(A).

Every single-outcome probability is at least zero, so adding probabilities for an event cannot produce a negative result.

The event uses only some outcomes from Ω, while all outcome probabilities together equal one; therefore the event probability cannot exceed one.

A and its complement Aᶜ contain every outcome exactly once between them, so P(A)+P(Aᶜ)=1.

A loaded six-sided die is modeled as follows: faces 1, 2, 3, and 4 each have probability 1/8; faces 5 and 6 each have probability 2/8=1/4. Find the probability of rolling an even number.

  1. The complete outcome list is Ω={1,2,3,4,5,6}. The event “even” contains exactly the outcomes {2,4,6}.
  2. Read the six assigned probabilities in face order: P(1)=1/8, P(2)=1/8, P(3)=1/8, P(4)=1/8, P(5)=2/8, and P(6)=2/8. Therefore the required probabilities are P(2), P(4), and P(6).
  3. Add 1/8+1/8+2/8=4/8=1/2. As a check, the odd outcomes have probability 1/8+1/8+2/8=1/2, and the two complementary probabilities add to one.

Result: P(even)=1/2.