Binomial Law Course

Lesson 1 • Conditional Probability

Defines probability given partial information and derives the multiplication rule. Directly prepares learners for independence, a key binomial assumption.

Lesson 2 • Probability Axioms and Rules

Introduces the three Kolmogorov axioms and derives addition rules. Provides the formal backbone for all probability calculations in the course.

Lesson 3 • Statistical Independence

Formalises independence between events and extends it to multiple trials. This concept is the central assumption underlying the binomial model.

Lesson 1 • Fundamental Counting Principle

Introduces the multiplication rule for sequential choices. Provides the base logic for all more advanced counting formulas used later.

Lesson 2 • Combinations and Binomial Coefficients

Derives the combination formula and connects it to binomial coefficients. These coefficients are the core counting tool inside the binomial formula.

Lesson 3 • Permutations

Counts ordered arrangements of objects with and without repetition. Distinguishes ordered from unordered selection, setting up combinations.

Lesson 1 • Probability Mass Functions

Specifies how probability is distributed across discrete values. Learners learn to construct, verify, and graph PMFs for simple experiments.

Lesson 2 • Common Discrete Distributions

Surveys Bernoulli, geometric, and Poisson distributions as context. Positions the binomial as one member of a broader discrete distribution family.

Lesson 3 • Variance and Standard Deviation

Quantifies spread around the mean using variance and its square root. These measures are computed explicitly for the binomial distribution later.

Lesson 4 • Random Variable Concept

Defines a random variable as a numerical function on a sample space. Bridges the abstract probability framework to measurable, computable quantities.

Lesson 1 • Assumptions of Repeated Bernoulli Trials

States the four conditions required for a valid sequence of Bernoulli trials. Verifying these assumptions is essential before applying the binomial law.

Lesson 2 • Binary Outcomes and Success Probability

Formalises trials with exactly two outcomes labelled success and failure. Establishes the parameter p as the probability of success in one trial.

Lesson 3 • Bernoulli Distribution Properties

Derives the PMF, mean, and variance of a single Bernoulli trial. These results are directly extended to n trials in the binomial model.

Lesson 1 • Parameters n and p

Explores how changing n and p reshapes the distribution. Builds intuition for parameter effects before formal statistical estimation.

Lesson 2 • Counting Successes in n Trials

Uses combinations to count sequences with exactly k successes. Connects Chapter 2 combinatorics directly to the binomial probability formula.

Lesson 3 • The Binomial Probability Formula

States P(X=k) = C(n,k) p^k q^(n-k) and interprets each component. Learners verify the formula sums to one using the binomial theorem.

Lesson 4 • Mean and Variance of the Binomial

Derives E[X] = np and Var(X) = npq using linearity of expectation. Provides closed-form results students apply in all subsequent chapters.

Lesson 1 • Case Studies Across Domains

Solves binomial problems from manufacturing, medicine, and finance contexts. Demonstrates the law's broad applicability and reinforces parameter identification.

Lesson 2 • Exact Probability Calculations

Applies the binomial formula to compute P(X=k) for specific scenarios. Reinforces formula mechanics through varied numerical examples.

Lesson 3 • Applied Problem-Solving Framework

Provides a structured approach: identify parameters, state the event, compute, interpret. Develops transferable problem-solving habits for any binomial scenario.

Lesson 4 • Using Statistical Software and Tables

Demonstrates binomial calculations using spreadsheet functions and statistical tables. Efficiency tools free cognitive load for interpretation rather than arithmetic.

Lesson 1 • Negative Binomial and Hypergeometric Connections

Introduces distributions that relax binomial assumptions for richer modelling. Clarifies boundaries of the binomial model by contrast with related laws.

Lesson 2 • Choosing the Right Approximation

Provides decision criteria for selecting exact, normal, or Poisson computation. Builds practical judgement that prevents misapplication in real analyses.

Lesson 3 • Poisson Approximation to the Binomial

Derives the Poisson limit when n is large and p is small. Identifies rare-event scenarios where this approximation is more accurate than normal.

Lesson 1 • Exact Binomial Test

Computes the exact p-value using cumulative binomial probabilities. Applies when sample sizes are small and normal approximation is unreliable.

Lesson 2 • Confidence Intervals for Proportions

Constructs exact and approximate confidence intervals for the parameter p. Complements hypothesis testing by quantifying estimation uncertainty.

Lesson 3 • Foundations of Hypothesis Testing

Defines null and alternative hypotheses, significance level, and p-value. Establishes the inferential framework applied to binomial data throughout this chapter.

Lesson 4 • Comparing Two Proportions

Extends binomial inference to test equality of two independent proportions. Covers pooled and unpooled test statistics for two-sample scenarios.