Event

Event

A set of possible outcomes resulting from a particular experiment. For example, a possible event when a single six-sided die is rolled is {5, 6}. That is, the roll could be a 5 or a 6.

In general, an event is any subset of a sample space (including the possibility of an the empty set).

See also

Probability

Key Formula

P(E) = \frac{\text{number of outcomes in } E}{\text{number of outcomes in } S}

Where:

$E$ = The event (a subset of the sample space)
$S$ = The sample space (the set of all possible outcomes)
$P(E)$ = The probability of event E occurring (valid when all outcomes are equally likely)

Worked Example

Problem: A single six-sided die is rolled. Let event E be "rolling a number greater than 4." List the outcomes in E and find its probability.

Step 1: Write the sample space. A standard die has six faces.

S = \{1, 2, 3, 4, 5, 6\}

Step 2: Identify which outcomes satisfy "greater than 4."

E = \{5, 6\}

Step 3: Count the outcomes. There are 2 outcomes in E and 6 outcomes in S.

P(E) = \frac{2}{6} = \frac{1}{3}

Answer: Event E = {5, 6}, and P(E) = 1/3 ≈ 0.333.

Another Example

Problem: Two coins are flipped. Let event A be "getting at least one head." List the outcomes in A and find its probability.

Step 1: Write the sample space for flipping two coins, where H = heads and T = tails.

S = \{HH, HT, TH, TT\}

Step 2: Pick out every outcome that contains at least one H.

A = \{HH, HT, TH\}

Step 3: There are 3 favorable outcomes out of 4 total.

P(A) = \frac{3}{4}

Answer: Event A = {HH, HT, TH}, and P(A) = 3/4 = 0.75.

Frequently Asked Questions

What is the difference between an event and an outcome in probability?

An outcome is a single possible result of an experiment (like rolling a 3). An event is a set of one or more outcomes grouped together by some condition (like "rolling an odd number" = {1, 3, 5}). Every individual outcome is technically an event too — a set with just one element — but most events contain multiple outcomes.

Can an event be the empty set?

Yes. The empty set ∅ is a valid event called the impossible event. It contains no outcomes, so its probability is 0. For example, "rolling a 7 on a standard die" corresponds to the empty set because no outcome in the sample space satisfies that condition.

Event vs. Outcome

An outcome is a single result of an experiment (e.g., rolling a 4). An event is a set that can contain zero, one, or many outcomes (e.g., "rolling an even number" = {2, 4, 6}). You calculate the probability of an event by counting how many of its outcomes are in the sample space. Think of an outcome as one element and an event as a subset — every outcome is a simple event, but not every event is a single outcome.

Why It Matters

Events are the building blocks of probability. Whenever you ask a question like "what is the chance of…", you are defining an event and then calculating its probability. Understanding events also lets you combine them using unions, intersections, and complements — the foundation for more advanced topics like conditional probability and independence.

Common Mistakes

Mistake: Confusing an event with a single outcome.

Correction: An event is a set of outcomes, written with curly braces. Even when an event has only one outcome, like {3}, it is still a set — not just the number 3. This distinction matters when you start combining events with set operations.

Mistake: Forgetting that the sample space itself and the empty set are both events.

Correction: The sample space S is the certain event (probability 1) and the empty set ∅ is the impossible event (probability 0). Both are valid subsets of S and therefore valid events.

Related Terms

Sample Space — The set of all possible outcomes
Outcome — A single result within a sample space
Experiment — The process that produces outcomes
Probability — The numerical measure of an event's likelihood
Subset — An event is a subset of the sample space
Set — Events are sets of outcomes
Empty Set — The impossible event with probability zero