AP Statistics Unit 2: Probability, Random Variables, and Probability Distributions Notes

Introduction

Probability models translate a repeatable chance process into events, conditional relationships, random variables, distributions, and sampling behavior.

Begin with the sample space and mechanism. Rules such as addition, multiplication, and complements follow from how events overlap or condition one another; they are not interchangeable formulas.

Random variables attach numbers to outcomes. Their distributions support expected value and standard deviation, while binomial and normal models apply only when their defining conditions fit. Sampling distributions then describe statistics across repeated random samples.

Two categorical variables

joint and conditional proportions

Chance process

events and probability rules

Random variable

distribution, parameters, models

Repeated samples

sampling distributions and CLT

Current exam guidance: Aligned to the effective Fall 2026 AP Statistics framework, including its current calculator and inference expectations.

2.1

Tabular and Graphical Representations for the Distributions of Two Categorical Variables

How do two-way tables and related graphs show two categorical variables at once?

Short answer

Interior cells show joint counts, margins show one-variable totals, and conditional displays rescale within a chosen row or column so group patterns can be compared.

Know firstOne-variable frequency tables · Relative frequency

Read cells, margins, and conditions separately

The table records 100 units by region and outcome class. Interior cells such as North/class 1 are joint counts. Row and column totals are marginal counts. Dividing a row by its total describes the outcome distribution within that region; dividing a column by its total answers a different conditional question.

Mosaic area represents joint proportion, while segmented bars use equal total widths to compare conditional distributions. A useful graph keeps both variables named, makes the conditioning direction explicit, and exposes the underlying counts in an accessible table.

Counts before percentages

Region	Class 1	Class 2	Class 3	Total
North	18	12	10	40
South	9	21	30	60
Total	27	33	40	100

Name the denominator

Locate the interior joint count.
Use margins to check the grand total.
Choose row, column, or grand-total conditioning from the question.
Label a segmented bar or mosaic with the same denominator.
Compare patterns without implying causation.

Keep these three ideas

Interior cells are joint; margins are marginal.
Conditioning changes the denominator.
Graph areas or segments must reproduce the table.

Keep reading Unit 2: Probability, Random Variables, and Probability Distributions

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2.1 Tabular and Graphical Representations for the Distributions of Two Categorical VariablesRead

2.2 Summary Statistics for Two Categorical Variables

2.3 Estimating Probabilities Using Simulation

2.4 Introduction to Probability

2.5 Mutually Exclusive Events

2.6 Conditional Probability

2.7 Independent Events and Unions of Events

2.8 Introduction to Random Variables and Probability Distributions

2.9 Parameters of Random Variables

2.10 The Binomial Distribution

2.11 The Normal Distribution

2.12 Sampling Distributions and the Central Limit Theorem

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