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Mathematics

Probability Rules: And, Or, Not, and Conditional

A High School & College Primer

Probability is the section where a lot of students hit a wall. The formulas look similar, the notation is cryptic, and it's easy to mix up when to add versus when to multiply. If you have a statistics quiz, a discrete math exam, or an AP Stats test coming up and you're still fuzzy on exactly why P(A or B) has that extra term subtracted from it, this guide is for you.

**Probability Rules: And, Or, Not, and Conditional** is a focused, no-filler primer covering the four rules every intro stats and discrete math student needs. Starting from scratch with sample spaces and basic probability, it walks you through the complement (Not) rule, the addition rule for unions, the multiplication rule for intersections, and conditional probability — including how to read P(A|B) off a two-way table without memorizing a formula cold. The final section ties everything together through tree diagrams and a worked Bayes-style problem, naming the exact mistakes students make on exams.

This guide is written for high school students in grades 9–12 and early college students who need a clear explanation fast — not a 400-page textbook. Every concept is defined in plain language, every rule is shown with worked numbers, and common misconceptions are called out directly. Parents helping a student prep for an intro statistics course and tutors looking for a concise session reference will find it equally useful.

If you need to walk into your next probability exam with genuine confidence, start here.

What you'll learn
  • Compute probabilities of single events using the basic definition and complement rule.
  • Apply the addition rule for 'or' events, distinguishing mutually exclusive from overlapping events.
  • Apply the multiplication rule for 'and' events, distinguishing independent from dependent events.
  • Compute and interpret conditional probabilities using the formula P(A|B) = P(A and B)/P(B).
  • Use tree diagrams and two-way tables to organize multi-step probability problems.
What's inside
  1. 1. What Probability Is and the Not Rule
    Sets up sample spaces, events, and the basic definition of probability, then introduces the complement (Not) rule as the simplest of the four rules.
  2. 2. The Or Rule: Addition for Unions
    Covers the addition rule for P(A or B), including mutually exclusive events and the inclusion-exclusion correction for overlap.
  3. 3. The And Rule: Multiplication for Intersections
    Introduces independence and the multiplication rule for P(A and B), contrasting independent and dependent events with worked examples.
  4. 4. Conditional Probability
    Defines P(A|B), derives it from the multiplication rule, and shows how to read conditional probabilities off two-way tables.
  5. 5. Putting It Together: Trees, Tables, and Multi-Step Problems
    Combines all four rules through tree diagrams and two-way tables, with a worked Bayes-style problem and common pitfalls.
Published by Solid State Press
Probability Rules: And, Or, Not, and Conditional cover
TLDR STUDY GUIDES

Probability Rules: And, Or, Not, and Conditional

A High School & College Primer
Solid State Press

Probability Rules: And, Or, Not, and Conditional

A High School & College Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Who This Book Is For

If you are a high school student who needs a probability rules study guide before a quiz, a unit exam, or an AP Stats probability rules quick review session, this book is for you. It also works if you are in an intro stats or discrete math probability primer course at the college level and the textbook is moving faster than you can follow.

This primer covers the four rules you will see on every probability test: the complement rule and sample space explained from scratch, then unions, intersections, and conditional probability explained simply. Along the way you will see the addition and multiplication rule for statistics in plain language, with worked numbers at every step. About 15 pages, no filler.

Read straight through once — the sections build on each other. Work every example yourself before reading the solution, then use the problem set at the end as your and/or/not probability worksheet to confirm you have it. That is the whole plan.

Contents

  1. 1 What Probability Is and the Not Rule
  2. 2 The Or Rule: Addition for Unions
  3. 3 The And Rule: Multiplication for Intersections
  4. 4 Conditional Probability
  5. 5 Putting It Together: Trees, Tables, and Multi-Step Problems
Chapter 1

What Probability Is and the Not Rule

Every probability problem starts with the same question: what could happen? Before you can calculate a single number, you need a clear picture of all the possible outcomes.

A sample space is the complete list of all possible outcomes of a random process. Roll one six-sided die: the sample space is $\{1, 2, 3, 4, 5, 6\}$. Flip a coin: the sample space is $\{H, T\}$. Draw one card from a standard deck: the sample space has 52 outcomes, one for each card. The key word is complete — nothing that could happen is left off the list, and nothing appears twice.

An event is any collection of outcomes from the sample space — in other words, any subset you care about. "Roll an even number" is the event $\{2, 4, 6\}$. "Draw a heart" is the set of all 13 heart cards. An event can contain one outcome, several, or even all of them. A single-outcome event like "roll exactly a 3" is sometimes called a simple event.

Defining Probability

Probability is a number between 0 and 1 (inclusive) that measures how likely an event is. A probability of 0 means the event is impossible; a probability of 1 means it is certain. Everything else falls in between.

When every outcome in the sample space is equally likely to occur, the probability of an event $A$ is:

$P(A) = \frac{\text{number of outcomes in } A}{\text{total number of outcomes in the sample space}}$

This is the definition you will use for most intro-level problems involving dice, cards, coins, and similar setups. It is clean and mechanical: count the favorable outcomes, count the total outcomes, divide.

Example. A bag contains 3 red marbles, 5 blue marbles, and 2 green marbles (10 total). You draw one marble at random. What is the probability of drawing a blue marble?

Solution. The sample space has 10 equally likely outcomes. The event "blue marble" contains 5 outcomes. $P(\text{blue}) = \frac{5}{10} = 0.5$ There is a 50% chance of drawing a blue marble.

Keep reading

You've read the first half of Chapter 1. The complete book covers 5 chapters in roughly fifteen pages — readable in one sitting.

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