Difficulty: Intermediate | Prerequisites: Lessons 1 – 2 (probability as long-run relative frequency, simulation basics).
Once you know what probability means, you need the rules for calculating it. This section gives you the toolkit: sample spaces, probability models, the complement rule, the addition rule for "or" probabilities, and the use of Venn diagrams and two-way tables to organise information. These rules are the backbone of every probability question on the AP exam, and they feed directly into the conditional probability and independence material that follows.
Every probability calculation starts with a sample space (the list of all possible outcomes) and a probability model (probabilities assigned to each outcome that are all between 0 and 1 and sum to 1). The complement rule, the addition rule for mutually exclusive events, and the general addition rule let you find probabilities of combined events. Venn diagrams and two-way tables are the main tools for organising overlapping events.
Event
Any collection of outcomes from a random process. Denoted by a capital letter (A, B, etc.).
In simple terms, an event is the specific thing you are asking about, like "drawing a king" or "rolling an even number."
Sample space
The set of all possible outcomes of a random process.
Think of it as the complete menu of everything that could happen. For a coin flip: {Heads, Tails}. For two coin flips: {HH, HT, TH, TT}.
Probability model
A description of a random process that lists the sample space and assigns a probability to each outcome.
In simple terms, it is the sample space plus the numbers that go with each outcome. The probabilities must all be between 0 and 1, and they must add up to exactly 1.
Empirical probability (experimental probability)
Probability estimated from observed data. You run the experiment and count.
Think of it as: "we tried it a bunch of times and here is what happened."
Theoretical probability
Probability calculated from a model using logical reasoning about equally likely outcomes or known structure.
Think of it as: "we worked it out on paper using the rules."
Complement
The complement of event A, written A^c, is the event that A does not occur. It consists of all outcomes in the sample space that are not in A.
In simple terms, "not A."
Complement rule
P(A^c) = 1 − P(A). The probability of something not happening is one minus the probability of it happening.
Mutually exclusive (disjoint) events
Two events that cannot occur at the same time. If one happens, the other cannot.
In simple terms, there is zero overlap. Drawing a King and drawing a Queen from a single card draw are mutually exclusive, because one card cannot be both.
Union (A ∪ B)
The event that A occurs, or B occurs, or both occur. Read "A or B."
Intersection (A ∩ B)
The event that both A and B occur simultaneously. Read "A and B."
General addition rule (general sum rule)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B). You subtract the intersection to avoid counting the overlap twice.
Think of it as: when two circles in a Venn diagram overlap, adding both circles counts the overlapping region twice, so you subtract it once.
Addition rule for mutually exclusive events
If A and B are mutually exclusive, P(A ∩ B) = 0, so the general rule simplifies to P(A ∪ B) = P(A) + P(B).
Venn diagram
A diagram using overlapping circles inside a rectangle to show the relationships between events. The rectangle represents the entire sample space.
List all outcomes in the sample space.
Assign a probability to each. Every probability must satisfy 0 ≤ P(outcome) ≤ 1.
All probabilities must sum to exactly 1.
Example (two fair coin flips): sample space = {HH, HT, TH, TT}, each with probability 0.25. Sum = 1. Valid model.
Sum the probabilities. If they do not equal 1, the model is invalid.
Check that no individual probability is negative or greater than 1.
Example (M&M colours): Blue 0.207, Orange 0.205, Green 0.198, Yellow 0.135, Red 0.131, Brown 0.124. Sum = 1.000. Valid model.
Useful when finding P(A) directly is hard, but finding P(A^c) is easy.
P(A) = 1 − P(A^c).
Example: probability of getting at least one head in 3 flips. Easier to calculate P(no heads) = P(TTT) = 0.125, then P(at least one head) = 1 − 0.125 = 0.875.
If A and B cannot happen together, P(A ∩ B) = 0.
P(A ∪ B) = P(A) + P(B).
Example: drawing one M&M. P(orange or blue) = 0.205 + 0.207 = 0.412, because one candy cannot be two colours at once.
When A and B can both happen at the same time, use P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Example: draw one card from a standard deck. P(black card or ace) = P(black) + P(ace) − P(black ace) = 26/52 + 4/52 − 2/52 = 28/52 ≈ 0.538.
Draw a rectangle for the sample space.
Draw overlapping circles for each event.
Fill in the intersection first, then work outward to the "only A" and "only B" regions, then the region outside both circles.
The four regions of a two-event Venn diagram: A only, B only, A ∩ B, and neither.
Always start by placing the intersection value, because you need it to calculate the other regions.
Rows and columns represent the categories of two variables. Cell counts give the joint frequencies.
Marginal totals (the row and column totals) give the individual event frequencies.
To find P(A ∩ B), read the cell where A's row meets B's column and divide by the grand total.
To find P(A ∪ B), use the general addition rule, or add the relevant cells carefully (avoiding double-counting).
Eats a Taco | Does Not Eat a Taco | Total | |
|---|---|---|---|
Watched Squid Games | 21 | 40 | 61 |
Did Not Watch | 29 | 10 | 39 |
Total | 50 | 50 | 100 |
Let E = "Eats a Taco" and S = "Watched Squid Games."
P(E) = 50/100 = 0.50
P(E ∩ S) = 21/100 = 0.21
P(E ∩ S^c) = 29/100 = 0.29
P(S^c) = 39/100 = 0.39
P(E^c ∪ S) = P(E^c) + P(S) − P(E^c ∩ S) = 50/100 + 61/100 − 40/100 = 71/100 = 0.71
Complement rule: P(A^c) = 1 − P(A)
Addition rule for mutually exclusive events: P(A ∪ B) = P(A) + P(B) [only when P(A ∩ B) = 0]
General addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Notation reminder: On the AP formula sheet, union (∪) replaces the word "or" and intersection (∩) replaces "and." A^c (or with a superscript C) means "not A."
Medical screening uses the addition rule constantly. If a patient could have disease A or disease B (or both), the probability of testing positive for at least one involves exactly this kind of "or" calculation with overlap. Quality control in manufacturing uses complements: it is easier to calculate the probability that a product has zero defects and subtract from 1 than to list every possible way it could be defective.
Students often add probabilities for "or" questions without subtracting the overlap. If the events can happen together, you must subtract P(A ∩ B). Only skip that step when the events are mutually exclusive.
Students sometimes think "mutually exclusive" and "independent" mean the same thing. They are different concepts and, in fact, mutually exclusive events (with non-zero probabilities) are never independent. This distinction is tested frequently.
Students forget that the region outside both circles in a Venn diagram still has probability. Always account for "neither A nor B."
When building probability models, students sometimes list probabilities that do not sum to 1. Always check the sum.
⚠️ The AP exam uses set notation (∪, ∩, superscript C) rather than the words "or," "and," and "not." Make sure you can read and write in both.
⚠️ Free-response questions involving two-way tables almost always require at least one union or intersection calculation. Show every step, including the formula and the substitution.
⚠️ Venn diagram questions require you to fill in all four regions. Partial diagrams lose marks.
⚠️ A probability model validity check (do the probabilities sum to 1, and are they all between 0 and 1?) is a common short-answer question.
True or false: If P(A) = 0.4 and P(B) = 0.5, then P(A ∪ B) must equal 0.9.
Fill in the blank: Events that cannot occur at the same time are called __________ events.
True or false: P(A^c) + P(A) = 1 for any event A.
Fill in the blank: In a valid probability model, all probabilities must be between and , and must sum to __.
True or false: In a Venn diagram with two overlapping events, the total probability inside both circles equals P(A) + P(B).
Answers: 1. False (only true if A and B are mutually exclusive; otherwise you must subtract the intersection). 2. Mutually exclusive (or disjoint). 3. True. 4. 0 and 1; 1. 5. False (it equals P(A) + P(B) − P(A ∩ B), because the overlap is counted twice).
Q: Suppose P(A) = 0.35, P(B) = 0.50, and P(A ∩ B) = 0.15. Find P(A ∪ B).
A: P(A ∪ B) = 0.35 + 0.50 − 0.15 = 0.70.
Q: You draw one card from a standard 52-card deck. What is P(King ∩ Queen)?
A: 0. A single card cannot be both a King and a Queen. These events are mutually exclusive.
Q: Using the M&M probability model (Blue 0.207, Orange 0.205, Green 0.198, Yellow 0.135, Red 0.131, Brown 0.124), find the probability that a randomly selected M&M is not green.
A: P(not green) = 1 − P(green) = 1 − 0.198 = 0.802.
Q: In the Squid Games / Taco two-way table, find P(E^c ∩ S), the probability a randomly selected person did not eat a taco and watched Squid Games.
A: 40/100 = 0.40.
Q: Are "eats a taco" and "watched Squid Games" mutually exclusive? Explain.
A: No. 21 people both ate a taco and watched Squid Games, so P(E ∩ S) = 0.21 ≠ 0. The events can occur together.
The complement rule and addition rule appear again in the binomial and geometric distribution formulas (Unit 6). Venn diagrams and two-way tables are the foundation for conditional probability (Lesson 5), which in turn is the basis for independence, a concept central to inference procedures in Units 8 – 9.
Related Terms / Search Tags: probability rules, sample space, probability model, complement rule, addition rule, general sum rule, mutually exclusive, disjoint events, union, intersection, Venn diagram, two-way table, AP Statistics Unit 5, P(A or B), P(A and B), set notation, overlapping events