Sample Spaces & Events
The vocabulary every probability rule is secretly built from: outcomes, events, and the set operations that combine them.
Start with the list of everything that could happen
Probability rules can feel like a bag of formulas until you have the right vocabulary. Underneath, every one of them is really about sets of outcomes — so we build that language first.
Run any experiment — flip a coin, roll a die, draw a card. Before you ask how likely anything is, write down the plain list of everything that could possibly happen. Rolling one die, that list is \(\{1,2,3,4,5,6\}\). That complete list is the star of this lesson.
- The sample space (written \(S\)) is the set of all possible outcomes of an experiment.
- An outcome is one single result in that list — one member of \(S\).
- An event is any collection of outcomes you care about — that is, a subset of the sample space.
So an event is not a mysterious new object: it is just part of the list you already wrote down.
An event is a subset you name
Roll one die, so \(S = \{1,2,3,4,5,6\}\). Here are three events, each just a subset:
- \(A = \{2,4,6\}\) — the roll is even.
- \(B = \{4,5,6\}\) — the roll is bigger than 3.
- \(C = \{6\}\) — you roll a six.
When every outcome is equally likely, the probability of an event is simply the fraction of the list it covers: \(P(A) = 3/6 = 1/2\). The event happens whenever the actual outcome is one of its members.
Combining events: not, and, or
Because events are sets, we combine them with three set operations. A Venn diagram — a box for the sample space with a circle for each event — makes all three easy to see:
- The complement \(A^{c}\) (not A) is every outcome outside \(A\). For the even event, \(A^{c} = \{1,3,5\}\).
- The intersection \(A \cap B\) (A and B) is the outcomes in both circles — where they overlap.
- The union \(A \cup B\) (A or B) is the outcomes in either circle, the overlap included.
Everyday or often means one or the other but not both. In probability, A or B means at least one of them — A alone, B alone, or both together. That is exactly why the overlap counts as part of the union.
Two events are mutually exclusive — also called disjoint — when they share no outcomes, so \(A \cap B = \varnothing\) and \(P(A \cap B) = 0\). Their circles do not overlap. Rolling a 2 and rolling a 5 on one die cannot both happen, so those events are disjoint.
With no overlap to double-count, the addition rule loses its last term: \(P(A \cup B) = P(A) + P(B)\). In the sim, drag the overlap to 0 to watch this happen.
- List the sample space and events: \(S = \{1,2,3,4,5,6\}\), \(A = \{2,4,6\}\), \(B = \{4,5,6\}\). Each outcome has probability \(1/6\), so \(P(A) = 3/6 = 1/2\) and \(P(B) = 3/6 = 1/2\).
- Intersection (and): the outcomes in both A and B are \(\{4,6\}\), so \(P(A \cap B) = 2/6 = 1/3\).
- Union (or): apply the addition rule \(P(A \cup B) = 1/2 + 1/2 - 1/3 = 2/3\). Check by listing: \(A \cup B = \{2,4,5,6\}\), which is \(4/6 = 2/3\).
- Complement (not A): \(P(A^{c}) = 1 - P(A) = 1 - 1/2 = 1/2\), matching the odd outcomes \(\{1,3,5\}\).
Check your understanding
- The sample space S is the set of all possible outcomes; a single result is an outcome.
- An event is a subset of the sample space — a named collection of outcomes.
- Combine events with set operations: complement (not A), intersection (A and B), union (A or B).
- Complement rule: P(not A) = 1 − P(A). Addition rule: P(A or B) = P(A) + P(B) − P(A and B).
- Mutually exclusive (disjoint) events share no outcomes, so P(A and B) = 0 and P(A or B) = P(A) + P(B).