Probability Rules: Complete Guide with Examples
Five rules govern all of probability theory. Master these and you can solve virtually any basic probability problem: the complement rule, addition rule, multiplication rule, conditional probability, and Bayes' theorem.
All probabilities are between 0 (impossible) and 1 (certain). The sum of all probabilities in a sample space equals exactly 1.
Rule 1: The Complement Rule
P(A) + P(not A) = 1, so P(not A) = 1 − P(A).
Use when it's easier to count what does NOT happen. Example: What is the probability of rolling at least one 6 in 4 rolls of a d6? P(no 6 in 4 rolls) = (5/6)⁴ = 625/1296 ≈ 48.2%. P(at least one 6) = 1 − 625/1296 = 671/1296 ≈ 51.8%.
The complement is the "at least one" trick: P(at least one) = 1 − P(none). This is far simpler than adding P(exactly one) + P(exactly two) + …
Rule 2: The Addition Rule
P(A or B) = P(A) + P(B) − P(A and B).
Subtract P(A and B) to avoid double-counting events that satisfy both A and B.
For mutually exclusive events (cannot both occur): P(A or B) = P(A) + P(B). Example: rolling a 2 or 5 on a d6 — mutually exclusive: P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3.
Non-mutually exclusive example: P(Ace or Heart) = P(Ace) + P(Heart) − P(Ace of Hearts) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13.
Rule 3: The Multiplication Rule
P(A and B) = P(A) × P(B|A) — the general multiplication rule.
For independent events: P(A and B) = P(A) × P(B). Events are independent if knowing A occurred tells you nothing about B.
Example (independent): P(two heads in a row) = P(H) × P(H) = 0.5 × 0.5 = 0.25.
Example (dependent): P(2 aces from a deck without replacement) = (4/52) × (3/51) = 12/2652 ≈ 0.45%.
KEY TEST for independence: P(A and B) = P(A) × P(B). If this holds, A and B are independent.
Rule 4: Conditional Probability
P(A|B) = P(A and B) / P(B). The probability of A given that B has occurred.
Conditional probability "restricts the sample space" to only outcomes where B occurred.
Example: A bag has 3 red and 2 blue marbles. P(red on 2nd draw | red on 1st, no replacement) = 2/4 = 0.5. The first draw changes the composition.
Rule 5: Bayes' Theorem
P(H|E) = P(E|H) × P(H) / P(E). Updates probability of hypothesis H given evidence E.
P(E) = P(E|H) × P(H) + P(E|not H) × P(not H) — the law of total probability.
Application: medical testing, spam filtering, search engine ranking, and machine learning all rely on Bayes' theorem to update beliefs from data.
Quick Reference Table
All five rules summarised with their formulas and when to use them:
| Rule | Formula | When to Use |
|---|---|---|
| Complement | P(not A) = 1 − P(A) | "At least one" problems |
| Addition (general) | P(A∪B) = P(A) + P(B) − P(A∩B) | Either event can occur |
| Addition (mutually exclusive) | P(A∪B) = P(A) + P(B) | Events cannot both occur |
| Multiplication (independent) | P(A∩B) = P(A) × P(B) | Both events, no dependence |
| Multiplication (dependent) | P(A∩B) = P(A) × P(B|A) | Both events, dependent |
| Conditional probability | P(A|B) = P(A∩B) / P(B) | Event A given B occurred |
| Bayes' theorem | P(H|E) = P(E|H)P(H)/P(E) | Update belief from evidence |