What Are the Rules of Probability? Every probability you will ever compute — whether for a coin flip, a card game, a machine failure rate, or a clinical trial — is governed by the same six rules. They are not arbitrary conventions. They follow logically from a single axiom: the set of all possible outcomes (the sample space) must have total probability exactly equal to 1. According to the foundational axioms of probability theory first formalized by the Soviet mathematician Andrei Kolmogorov in 1933, and as presented in standard statistics textbooks such as OpenStax Introductory Statistics (available free at openstax.org ), the entire structure of modern probability rests on three axioms: non-negativity, unit measure (total probability = 1), and countable additivity. The six rules covered here are direct consequences. The 6 Core Probability Rules The following six rules cover every fundamental operation in probability. Each one has a specific activation condition — the situation that determines whether it applies. Learn the conditions, and you will always know which rule to use. Rule 1: The Range Rule (Probability Boundaries) The Range Rule is the gatekeeper of all probability. It says that every probability must be a number between 0 and 1, inclusive. A probability of 0 means the event is impossible — it will never happen under any circumstance within the defined experiment. A probability of 1 means the event is certain — it will happen every single time. Any probability in between represents a partial likelihood. If your calculated probability is negative or greater than 1, you have made an error. The most common cause is forgetting to subtract P(A ∩ B) in the General Addition Rule for overlapping events, causing the result to exceed 1. Go back and recheck your inputs and formula choice. Rule 2: The Complement Rule The Complement Rule is one of the most useful shortcuts in probability. Since every outcome in the sample space either belongs to event A or does not belong to event A — with no overlap and nothing left out — the probabilities P(A) and P(A') must sum to exactly 1. Therefore, if you know one, you know the other. The complement is especially powerful when computing "at least one" problems. Instead of listing every way at least one success can occur, you compute 1 minus the probability of zero successes, which is usually a single calculation. Problem: A bag contains 3 red and 7 blue marbles. What is the probability of NOT drawing red? Find P(red). There are 3 red marbles out of 10 total: P(red) = 3/10 = 0.3 Apply the Complement Rule. P(not red) = 1 − P(red) = 1 − 0.3 = 0.7 ✓ P(not red) = 0.7 (70%). This matches the direct count: 7 blue marbles / 10 total = 7/10 = 0.7 ✓ Rule 3: The Addition Rule (Mutually Exclusive Events) Two events are mutually exclusive (also called disjoint) when they share no outcomes — they cannot occur simultaneously in a single trial. On a single roll of a die, rolling a 3 and rolling a 5 are mutually exclusive: you cannot get both on one roll. For such events, the addition rule is simple: just add the probabilities directly. Problem: What is the probability of rolling a 3 OR a 5 on a single fair six-sided die? Identify the events. A = roll a 3, B = roll a 5. Both have 1 favorable outcome out of 6 total. P(A) = 1/6, P(B) = 1/6. Check for mutual exclusivity. On one roll, can you get both 3 AND 5? No — a single die shows exactly one face. Therefore A and B are mutually exclusive: P(A ∩ B) = 0. Apply Rule 3. P(3 or 5) = P(3) + P(5) = 1/6 + 1/6 = 2/6 = 1/3 ✓ P(rolling a 3 or a 5) = 2/6 = 1/3 ≈ 0.333 (33.3%). Direct verification: {3, 5} are 2 favorable outcomes out of 6 total = 2/6 ✓ Rule 4: The General Addition Rule (Overlapping Events) When events can co-occur — when the outcome of one event is compatible with the outcome of another — simply adding P(A) + P(B) counts their shared outcomes twice. The General Addition Rule corrects this by subtracting P(A ∩ B), the probability of both events happening at the same time. This is the formula to use any time the word "OR" connects two events that are not mutually exclusive. Using P(A ∪ B) = P(A) + P(B) when the events overlap is the single most common error on probability exams. Always ask: "Can both events occur at the same time?" If the answer is yes, you must subtract P(A ∩ B). Skipping this step overcounts the shared outcomes and gives an answer greater than the true probability — sometimes even exceeding 1. Problem: What is the probability of drawing a King OR a Red card from a standard 52-card deck? Identify the events and their counts. A = King: 4 Kings in 52 cards. B = Red card: 26 red cards (13 hearts + 13 diamonds) in 52 cards. Find the intersection. A ∩ B = King AND Red = Red Kings. There are exactly 2 Red Kings (King of Hearts, King of Diamonds). P(A ∩ B) = 2/52. Apply Rule 4. P(King or Red) = P(King) + P(Red) − P(King and Red) = 4/52 + 26/52 − 2/52 = 28/52 = 7/13 ✓ P(King or Red card) = 28/52 = 7/13 ≈ 0.538 (53.8%). Direct count: 26 red cards + 2 black kings (non-red kings) = 28 favorable cards ✓ Rule 5: The Multiplication Rule (Independent Events) Two events are independent when knowing the outcome of one provides no information about the other. Flipping a coin and rolling a die are independent: heads or tails on the coin has no bearing on what number appears on the die. For independent events, multiply the individual probabilities. Problem: What is the probability of flipping Heads AND rolling a 6 simultaneously? Identify independence. The coin flip and the die roll are separate experiments. One cannot influence the other — they are independent events. Find individual probabilities. P(Heads) = 1/2. P(rolling a 6) = 1/6. Apply Rule 5. P(Heads AND 6) = P(Heads) × P(6) = 1/2 × 1/6 = 1/12 ✓ P(Heads and rolling 6) = 1/12 ≈ 0.0833 (8.33%). The sample space has 12 equally likely outcomes (2 coin × 6 die), and only 1 is (H, 6) ✓ Rule 6: The Multiplication Rule (Dependent Events) Events are dependent when the outcome of the first changes the probability of the second. The classic example is drawing cards without replacement: once the first card is removed from the deck, the total count drops from 52 to 51, and the probability of any specific card on the second draw changes accordingly. The notation P(B|A) — the conditional probability of B given A — captures this updated probability. Using P(A ∩ B) = P(A) × P(B) when drawing cards without replacement (or any "without replacement" scenario) is incorrect. After the first draw, the deck has 51 cards, not 52. Always update the probability for the second event based on what the first event removed. Problem: Drawing two Aces consecutively from a standard deck without replacement. First draw. There are 4 Aces in a 52-card deck. P(Ace on 1st draw) = 4/52 = 1/13. Update for the second draw. Since no replacement occurs, the deck now has 51 cards and only 3 remaining Aces. The conditional probability P(Ace on 2nd | Ace on 1st) = 3/51 = 1/17. Apply Rule 6. P(Ace, then Ace) = P(1st Ace) × P(2nd Ace | 1st Ace) = 4/52 × 3/51 = 12/2652 = 1/221 ✓ P(two consecutive Aces, no replacement) = 1/221 ≈ 0.00452 (0.452%). Compare the wrong answer using the independent rule: 4/52 × 4/52 = 16/2704 ≈ 0.59% — that overcounts by ~30% and is incorrect. The Ultimate Probability Rules Master Matrix This table is your complete reference for all six probability rules. Every rule in one place: name, formula, required condition, and plain-English meaning. Bookmark it, print it, and paste it into your notes. OR always triggers an Addition Rule (+). AND always triggers a Multiplication Rule (×). The next question is always: what kind of OR or AND? That determines which specific version of the rule to use. The "Add or Multiply?" Decision Tree Use this decision tree whenever you face a probability problem and are unsure which rule to apply. Start at the top, answer each question, and follow the branch to the exact rule you need. 🔀 Which Probability Rule Do I Use?

Can A and B happen at the same time? Does the first event affect the probability of the second? 💡 Tip: "Without replacement" problems are almost always dependent. "With replacement" problems are almost always independent. Visual Explanations: Venn Diagrams and Tree Diagrams Venn Diagrams: Visualizing Addition Rules Venn diagrams use overlapping circles to represent events within a sample space (the surrounding rectangle, labeled S). The area of a region corresponds to its probability. They are the clearest way to visualize the difference between mutually exclusive and overlapping events. No overlap — circles are separate. Rule 3 applies: P(A ∪ B) = P(A) + P(B) Overlap exists — subtract intersection. Rule 4 applies: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) Tree Diagrams: Visualizing Multiplication Rules A tree diagram maps every possible sequence of outcomes as branches of a tree. Each branch is labeled with its probability. To find the probability of a complete path (a sequence of outcomes), multiply the probabilities along that path. Tree diagrams make the Multiplication Rule visual and intuitive. Each path = multiply probabilities along that branch. P(H, 6) = 1/2 × 1/6 = 1/12 (starred path above). This is the Multiplication Rule for Independent Events in visual form. Probability Rules Calculator Use the interactive calculator below to apply any of the four operational probability rules to your own numbers. Select the rule tab, enter your values, and click Calculate. Probability Rules Calculator Enter P(A) to find P(A') = 1 − P(A) General Addition Rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, enter 0 for P(A ∩ B). For independent events: P(A ∩ B) = P(A) × P(B). For dependent events: enter P(B|A) in the second field. Critical Pitfalls: The Most Common Probability Mistakes These are the errors that cost students marks on exams. Each one stems from applying a rule without checking its required condition first. After every calculation, verify your answer is between 0 and 1. If P(A ∪ B) comes out greater than 1, you almost certainly forgot to subtract P(A ∩ B). If it is negative, check that P(A ∩ B) is not larger than either P(A) or P(B) — the intersection can never exceed either individual event's probability. Key Vocabulary: Probability Rules Glossary These are the precise definitions every probability rule depends on. Understanding these terms is prerequisite to applying the rules correctly. Probability Rules Quick Reference Cheat Sheet This is a condensed cheat sheet for all six rules. Print this section or bookmark this page for fast reference during homework, exams, or study sessions. The formulas below match standard textbook notation as used by Khan Academy, OpenStax, and Wolfram MathWorld. All probabilities fall between 0 and 1 inclusive. P = 0 is impossible; P = 1 is certain. Complement Rule Probability of "not A" = 1 minus probability of A. P(A) + P(A') = 1 always. Addition (Mutually Exclusive) OR of events that cannot both happen. P(A ∩ B) = 0. General Addition (Overlapping) OR of events that can both happen. Subtract overlap to avoid double-counting. Multiplication (Independent) AND of events where one does not affect the other. Multiply directly. Multiplication (Dependent) AND of events where first changes the second. Use conditional probability P(B|A). Range: 0 ≤ P(A) ≤ 1 — never below 0, never above 1 Complement: P(A') = 1 − P(A) — probability of "not A" OR + Mutually Exclusive: P(A ∪ B) = P(A) + P(B) OR + Overlapping: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) AND + Independent: P(A ∩ B) = P(A) × P(B) AND + Dependent: P(A ∩ B) = P(A) × P(B|A) Memory anchor: OR = Add | AND = Multiply Conditional probability: P(B|A) = P(A ∩ B) / P(A) Build On These Rules: Next Topics to Study Mastering the six probability rules opens the door to more advanced topics. The rules covered here are prerequisites for everything below. Next Steps in Probability Where These Rules Lead Deepens your understanding of P(B|A) and dependent events. Required for Bayes' Theorem. Uses the dependent multiplication rule to update probabilities in light of new evidence. The foundational page that establishes P(A) = favorable/total before the rules are applied. For additional worked problems aligned to standardized exam format, the Khan Academy Probability Library offers free, structured practice. For rigorous mathematical definitions, Wolfram MathWorld's Probability section provides formal notation consistent with academic coursework. The OpenStax Introductory Statistics textbook Chapter 3 covers these rules in a peer-reviewed, freely available format cited by university courses worldwide.