Basic Probability Rules
Probability is the mathematical language we use to describe how likely events are to occur. Ranging from 0 (impossible) to 1 (certain), a probability measures the proportion of outcomes that correspond to a specific event within a well-defined sample space. Whether you are predicting the chance of rain, analyzing the fairness of a game, or modeling genetic inheritance, the same foundational rules apply. The sample space S is the complete set of all possible outcomes of an experiment. An event A is any subset of S. The basic probability rules — including the complement rule, the addition rule, and the multiplication rule — allow us to calculate probabilities for complex events by breaking them into simpler pieces. These rules are not mere formulas to memorize; they follow logically from a small set of axioms first formalized by Andrey Kolmogorov in 1933. In this article you will learn how to assign probabilities to simple events, how to use set-theoretic ideas like union and intersection, and how to apply the four core rules systematically. Mastering these rules forms the foundation for every more advanced topic in probability, from conditional probability and Bayes' theorem to random variables and statistical inference.
- Rule 1 — Non-negativity and Bounds
- Rule 2 — The Sum of All Outcomes
- Rule 3 — The Complement Rule
- Rule 4 — The Addition Rule
Rule 1 — Non-negativity and Bounds
For any event A in sample space S, the probability satisfies 0 ≤ P(A) ≤ 1. A probability of 0 means the event cannot happen; a probability of 1 means it is certain to happen. These bounds are not a convention — they follow from the first Kolmogorov axiom. When you calculate a probability and get a negative number or a number greater than 1, you have made an error.
Rule 2 — The Sum of All Outcomes
The probabilities of all outcomes in the sample space must sum to exactly 1: P(S) = 1. For a fair six-sided die, each face has probability 1/6, and 6 × (1/6) = 1. For any discrete sample space {o₁, o₂, …, oₙ}, this means P(o₁) + P(o₂) + ⋯ + P(oₙ) = 1. This rule ensures our probability model accounts for every possibility.
Rule 3 — The Complement Rule
The complement of event A, written A' (or Aᶜ), is the set of all outcomes in S that are NOT in A. Because A and A' together cover the entire sample space without overlap, P(A) + P(A') = 1, which gives P(A') = 1 − P(A). This rule is extremely useful: if a direct calculation of P(A) is hard, try computing 1 − P(A') instead. For example, the probability of getting at least one head in three coin flips equals 1 − P(no heads) = 1 − (1/2)³ = 7/8.
Rule 4 — The Addition Rule
For any two events A and B, the probability that at least one of them occurs is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The subtraction removes the double-counting of outcomes that fall in both events. When A and B are mutually exclusive (they cannot both occur), P(A ∩ B) = 0, so the rule simplifies to P(A ∪ B) = P(A) + P(B). For example, a single card drawn from a standard deck: P(King or Heart) = P(King) + P(Heart) − P(King of Hearts) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13.
The Key Equation
- P(A \cup B) — Probability of A union B (dimensionless): The probability that event A occurs, event B occurs, or both occur.
- P(A) — Probability of A (dimensionless): The probability that event A occurs on its own.
- P(B) — Probability of B (dimensionless): The probability that event B occurs on its own.
- P(A \cap B) — Probability of A intersect B (dimensionless): The probability that both A and B occur simultaneously; subtracted to avoid counting those outcomes twice.
Worked Example
Problem. A student randomly selects one card from a standard 52-card deck. Let A = the card is a red card, and let B = the card is a face card (Jack, Queen, or King). Find P(A ∪ B), the probability that the card is red or a face card.
Answer. The probability of selecting a red card or a face card is 8/13, approximately 61.5%.
Origins · 1 minAndrey Nikolaevich Kolmogorov and how this was discovered
In his 1933 monograph Grundbegriffe der Wahrscheinlichkeitsrechnung (Foundations of the Theory of Probability), Kolmogorov placed probability on a rigorous mathematical foundation using measure theory. His three axioms — non-negativity, normalization, and countable additivity — are still the standard basis for all of modern probability theory, including every rule covered in this article. Kolmogorov was also a pioneer in algorithmic complexity theory and turbulence mathematics. He supervised more than 60 doctoral students during his career at Moscow State University.
Related Concepts
- Conditional Probability — Conditional probability extends the basic rules by asking how the probability of A changes given that B is known to have occurred, introducing P(A|B) = P(A ∩ B) / P(B).
- Independent Events — Two events are independent when P(A ∩ B) = P(A)·P(B), a special case of the multiplication rule that simplifies many probability calculations.
- Sample Spaces and Events — A thorough understanding of how to construct sample spaces using lists, tables, and tree diagrams is essential before applying any probability rule.
- Bayes' Theorem — Bayes' theorem combines the addition rule, complement rule, and conditional probability to update probabilities as new information arrives.