Difficulty: Introductory | Prerequisites: Basic algebra, familiarity with set notation helpful but not required.
Probability is the branch of mathematics that deals with randomness and uncertainty. This chapter sits at the beginning of the inferential statistics arc: before you can draw conclusions from data, you need a formal language for describing what "could happen" and how likely it is. Everything here, from sample spaces to set operations, becomes the scaffolding for probability rules, distributions, and hypothesis testing later in the course. If you have not seen set notation (unions, intersections, complements) before, this is where it gets introduced.
Probability is the study of randomness. Experiments produce outcomes, the full collection of which is the sample space. Events are subsets of that space, and set theory (complement, union, intersection) gives us the language to combine and relate them. Venn diagrams make all of this visual.
Random experiment
An activity with at least two possible outcomes whose result cannot be predicted with absolute certainty. Tossing a coin, rolling a die, and drawing a card are classic examples. In simple terms, it is any process where you genuinely do not know what will happen next.
Outcome
The result of a single run of an experiment. If you roll a die, one outcome is "landed on 4." Think of it as one specific thing that can happen.
Trial
One performance of the experiment. Roll the die once, that is one trial. In simple terms, a trial is doing the experiment a single time.
Sample space (S or Ω)
The complete set of all possible outcomes of an experiment. Think of it as the master list: every single thing that could happen, written out.
Event (A, E)
A collection (subset) of outcomes from the sample space. In simple terms, an event is any group of outcomes you decide to lump together and give a name.
Simple event
An event that contains exactly one outcome. Think of it as the most granular thing that can happen, with no further breakdown.
Empty event (∅ or { })
An event with no outcomes in it. It cannot occur. In simple terms, it is the "nothing happened" set, useful mainly as a bookkeeping concept.
A random experiment must have at least two possible outcomes and its result must be uncertain.
Each run of the experiment is called a trial; each result is an outcome.
A tree diagram is a common way to list all outcomes when an experiment has multiple stages.
For example, flipping a coin three times: the first branch splits into S (success) and F (failure), each of those splits again, and again, yielding outcomes like SSS, SSF, SFS, SFF, FSS, FSF, FFS, FFF.
The total number of outcomes for n stages with k options per stage is k^n. For three coin flips: 2^3 = 8.
The sample space S contains every possible outcome.
An event is any subset of S. It can contain one outcome (simple event), several, all of them (the certain event, S itself), or none (the empty event, ∅).
When describing events, list the outcomes inside curly braces.
Example: if the sample space for one die roll is S = {1, 2, 3, 4, 5, 6}, the event "roll an even number" is A = {2, 4, 6}.
Understanding these three operations is essential. They are the building blocks of every probability rule that follows.
Complement (A' or Aᶜ)
All outcomes in the sample space that are not in A.
Key word: NOT.
If A = {2, 4, 6} in a die roll, then A' = {1, 3, 5}.
Union (A ∪ B)
All outcomes that are in A, in B, or in both.
Key word: OR.
If A = {1, 2} and B = {2, 3}, then A ∪ B = {1, 2, 3}.
Intersection (A ∩ B)
Only the outcomes that are in both A and B simultaneously.
Key word: AND.
If A = {1, 2} and B = {2, 3}, then A ∩ B = {2}.
Disjoint / mutually exclusive events
Two events are disjoint when they share no outcomes: A ∩ B = ∅.
Key word: DISJOINT or MUTUALLY EXCLUSIVE.
If A = "roll a 1" and B = "roll a 6," those are disjoint because no single roll can be both.
Venn diagrams give a visual shorthand for set operations. A rectangle represents the sample space Ω. Circles inside it represent events.
Complement (A'): the shaded region is everything inside the rectangle but outside circle A.
Union (A ∪ B): the shaded region covers the entirety of both circles, including their overlap.
Intersection (A ∩ B): only the overlapping zone of the two circles is shaded.
Disjoint events: the two circles do not touch at all, with no overlap region.
Tree diagrams and sample spaces are used in quality control (mapping every possible defect path on an assembly line), network reliability (listing all combinations of components that could fail), and game design (enumerating every possible hand or board state). Whenever an engineer or analyst needs to be certain they have accounted for every scenario, they start by writing out the sample space.
Students often confuse "mutually exclusive" with "independent." They are different concepts. Mutually exclusive means the events cannot both happen; independent means one happening does not change the probability of the other. Two events that are mutually exclusive and both have positive probability are never independent.
Students sometimes think the complement of an event is its "opposite" in a colloquial sense. It is not vague: A' is precisely the set of outcomes in S that are not in A, no more and no less.
Forgetting the empty set ∅ is a valid event. It simply has probability zero.
Mixing up union and intersection. Remember: union = OR (bigger set), intersection = AND (smaller or equal set).
⚠️ Tree diagrams are a favourite for counting total outcomes in multi-stage experiments. Be ready to draw one and list the sample space from it.
⚠️ Set operation keywords (NOT, OR, AND) appear directly in word problems. Translating English into the correct operation is half the battle.
⚠️ Disjoint vs independent is a classic exam distinction question. Know the definitions cold.
⚠️ Venn diagram shading questions are common. Practice identifying which region corresponds to expressions like (A ∪ B)', A' ∩ B, etc.
True or False: A sample space can contain an infinite number of outcomes.
Fill in the blank: The key word for union is ______.
True or False: If A and B are mutually exclusive, then A ∩ B = ∅.
Fill in the blank: The complement of event A contains all outcomes in ______ that are not in A.
True or False: A simple event contains two or more outcomes.
Answers: 1. True (e.g. measuring a continuous variable). 2. OR. 3. True. 4. The sample space (S). 5. False, it contains exactly one.
Q: A coin is flipped three times. Write out the full sample space.
A: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. There are 2^3 = 8 outcomes.
Q: Let S = {1, 2, 3, 4, 5, 6}. If A = {1, 3, 5} and B = {2, 4, 6}, are A and B disjoint? What is A ∪ B?
A: Yes, A ∩ B = ∅ so they are disjoint. A ∪ B = {1, 2, 3, 4, 5, 6} = S.
Q: Using the same sample space, let C = {1, 2, 3} and D = {3, 4, 5}. What is C ∩ D? What is C'?
A: C ∩ D = {3}. C' = {4, 5, 6}.
Q: Explain in one sentence why two disjoint events with positive probability cannot be independent.
A: If A and B are disjoint and A occurs, B definitely did not occur, so knowing A changes the probability of B from something positive to zero, which violates the definition of independence.
This material connects directly to the probability rules covered in Part 2 of these notes (complement rule, addition rule) because those rules are stated in terms of unions, intersections, and complements. It also feeds into conditional probability and Bayes' theorem (Part 3), where intersections appear in every formula. Later in the course, when you study random variables and distributions, the sample space is what defines the set of values a random variable can take.
probability, sample space, outcome, trial, experiment, event, simple event, empty set, null event, complement, union, intersection, disjoint, mutually exclusive, Venn diagram, set theory, tree diagram, STAT, PIV 203, Chapter 4, introduction to statistics, Purdue