Difficulty: Introductory | Prerequisites: Basic set notation (curly braces, element-of symbol). Comfortable with listing outcomes of simple experiments such as coin flips or dice rolls.
This material is the foundation for everything else in STAT 3021. Before you can calculate a probability, you need a way to describe what can happen (the sample space) and a language for combining and comparing events (set operations). Chapters on probability rules, conditional probability, and Bayes' theorem all assume you can fluently read and write set expressions. If you are coming in cold, think of this as learning the alphabet before writing sentences: it is brief but not optional.
A sample space lists every possible outcome of an experiment. Events are subsets of that space, and you combine or compare them using union, intersection, and complement. Two events are mutually exclusive when they share no outcomes, meaning they cannot both happen at once.
Sample space (S)
The set of all possible outcomes of a statistical experiment.
Think of it as the complete menu of things that could happen. Roll two dice? S is every ordered pair from (1,1) to (6,6).
Event
Any subset of the sample space.
In simple terms, an event is a collection of outcomes you care about, such as "the sum is 7" or "both dice show even."
Union (A ∪ B)
The set of all elements that belong to A, to B, or to both.
Think of it as "A or B (or both)." If A = {2, 4, 8} and C = {3, 4, 5}, then A ∪ C = {2, 3, 4, 5, 8}.
Intersection (A ∩ B)
The set of elements that belong to both A and B simultaneously.
Think of it as "A and B at the same time." Using the same sets, A ∩ C = {4}.
Complement (A′ or Aᶜ)
The set of all elements in the sample space S that are not in A.
In simple terms, everything that is not A. If S = {0,1,2,...,9} and C = {2,3,4,5}, then C′ = {0,1,6,7,8,9}.
Empty set (∅ or φ)
A set containing no elements.
This appears when two events have no outcomes in common. If A ∩ B = ∅, the events never happen together.
Mutually exclusive (disjoint) events
Two events A and B such that A ∩ B = ∅.
In simple terms, if one happens the other cannot. Rolling a 2 and rolling a 5 on a single die are mutually exclusive; rolling an even number and rolling a number greater than 3 are not (4 and 6 satisfy both).
Set-builder notation
A compact way of describing a set by stating the rule its elements satisfy, e.g. S = {(x, y) | 1 ≤ x, y ≤ 6}.
Think of it as a filter: "all pairs (x, y) where x and y are each between 1 and 6."
There are two standard ways to write a sample space.
Roster (list) method: Write out every outcome explicitly. For two dice: S = {(1,1), (1,2), ..., (6,6)}, giving 36 ordered pairs. Use this when the space is small enough to enumerate.
Set-builder method: State the rule. S = {(x, y) | 1 ≤ x, y ≤ 6}. Preferred when listing every element would be impractical or when the space is continuous.
Both forms describe the same set. The roster method is clearer for small experiments; set-builder notation scales to larger or infinite sample spaces.
Suppose S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {0, 2, 4, 6, 8}, B = {1, 3, 5, 7, 9}, C = {2, 3, 4, 5}, D = {1, 6, 7}.
A ∪ C = {0, 2, 3, 4, 5, 6, 8}. Collect every element that appears in A, in C, or in both.
A ∩ B = ∅. A holds the even digits, B the odd digits; they share nothing.
C′ = {0, 1, 6, 7, 8, 9}. Everything in S that is not in C.
Compound operation: (C′ ∩ D) ∪ B. First find C′ ∩ D = {1, 6, 7}. Then take the union with B = {1, 3, 5, 7, 9}, giving {1, 3, 5, 6, 7, 9}.
(S ∩ C)′ = C′. The intersection of S with any subset is that subset itself, so the complement gives C′ = {0, 1, 6, 7, 8, 9}.
A ∩ C ∩ D′. A ∩ C = {2, 4}. D′ = {0, 2, 3, 4, 5, 8, 9}. The final intersection is {2, 4}.
Two events are mutually exclusive when their intersection is empty. From the homework:
Q2.18(b): The events in part (b) are mutually exclusive, meaning they cannot occur at the same time.
Q2.18(c): The events in part (c) are not mutually exclusive, meaning they share at least one outcome.
The distinction matters because for mutually exclusive events, P(A ∪ B) simplifies to P(A) + P(B). If the events overlap, you must subtract the intersection to avoid double-counting.
A \cup B = \{x : x \in A \text{ or } x \in B\}A \cap B = \{x : x \in A \text{ and } x \in B\}A' = \{x : x \in S \text{ and } x \notin A\}De Morgan's Laws (used frequently in probability proofs):
(A \cup B)' = A' \cap B'(A \cap B)' = A' \cup B'In words: the complement of a union is the intersection of the complements, and vice versa.
Other useful identities:
A ∪ A′ = S (an event and its complement cover the whole sample space)
A ∩ A′ = ∅
(A′)′ = A
S ∩ A = A
∅ ∩ A = ∅
Sample spaces and set operations are the scaffolding behind quality control (defective vs non-defective items in a batch), medical testing (true positives, false positives, and all their overlaps), and network reliability (which component failures bring the system down). Every time you see a Venn diagram in a news article about overlapping voter demographics or drug interactions, you are looking at set operations in action.
Students often confuse "mutually exclusive" with "independent." They are different concepts. Mutually exclusive means A ∩ B = ∅. Independent means P(A ∩ B) = P(A)P(B). Two mutually exclusive events with non-zero probability are never independent.
Forgetting that union means "or" (inclusive), not "exclusive or." A ∪ B includes outcomes in both A and B.
Writing a complement relative to the wrong universal set. C′ depends on what S is. Always check what sample space you are working in.
Confusing ordered and unordered sample spaces. Rolling die 1 then die 2 gives ordered pairs, so (2, 5) and (5, 2) are distinct outcomes. If order does not matter, the sample space is different and smaller.
⚠️ Expect to be asked to list a sample space both in roster form and in set-builder notation, then identify specific events within it.
⚠️ Set operation questions often chain multiple steps (e.g. find C′ ∩ D, then take a union). Work inside out and write the intermediate result before moving to the next operation.
⚠️ "Mutually exclusive or not?" is a standard short-answer question. The answer is always about whether the intersection is empty. Give the intersection, not a vague justification.
True or false: If A ∩ B = ∅, then A and B are mutually exclusive.
Fill in the blank: The complement of C relative to S contains all elements in ______ that are not in ______.
True or false: (A ∪ B)′ = A′ ∪ B′.
If S = {1, 2, 3, 4, 5} and A = {1, 3, 5}, what is A′?
True or false: Two events can be both mutually exclusive and independent if both have non-zero probability.
Answers: 1. True. 2. S; C. 3. False (it equals A′ ∩ B′, by De Morgan's Law). 4. {2, 4}. 5. False.
Q: Two fair dice are rolled. Write the sample space using set-builder notation.
A: S = {(x, y) | 1 ≤ x ≤ 6, 1 ≤ y ≤ 6}. There are 36 elements.
Q: Given S = {0,1,2,...,9}, A = {0,2,4,6,8}, B = {1,3,5,7,9}, and C = {2,3,4,5}, find (A ∩ C) and A ∩ B.
A: A ∩ C = {2, 4}. A ∩ B = ∅ (no element is both even and odd).
Q: Using the same sets, find C′ and then (C′ ∩ D) ∪ B where D = {1,6,7}.
A: C′ = {0,1,6,7,8,9}. C′ ∩ D = {1,6,7}. (C′ ∩ D) ∪ B = {1,3,5,6,7,9}.
Q: Events E and F are defined on the same sample space. E ∩ F = ∅. Are they mutually exclusive? Explain.
A: Yes. Mutually exclusive means the events share no outcomes, which is exactly what A ∩ B = ∅ states.
Q: Are events "rolling a number greater than 3" and "rolling an even number" on a single fair die mutually exclusive?
A: No. The outcomes 4 and 6 satisfy both events, so their intersection is non-empty.
This material connects directly to the addition rule for probability (Chapter 2 continued): P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The subtraction term disappears precisely when the events are mutually exclusive.
Set operations also underpin conditional probability and Bayes' theorem. Conditioning on event B means restricting your sample space to B, and every calculation inside that restricted space uses the same intersection and complement mechanics.
Later in the course, you will use sample spaces for random variables. The sample space of a continuous random variable is an interval (or all of the real line), and events become ranges of values.
Sample space, outcome, event, subset, universal set, Venn diagram, union, intersection, complement, mutually exclusive, disjoint events, set-builder notation, roster method, De Morgan's laws, empty set, null event, exhaustive events, STAT 3021, Walpole Ch. 2, probability foundations, set theory for statistics