Events and Sample Space

Probability is the mathematical language for describing uncertainty. Whenever we perform a random experiment — rolling a die, flipping a coin, measuring tomorrow's rainfall — we cannot predict the exact outcome in advance. The foundation of all probability theory rests on two key ideas: the sample space, which lists every possible outcome, and events, which are the specific results we care about. Mastering these concepts unlocks the ability to calculate likelihoods, analyse data, and reason rigorously under uncertainty. The sample space is denoted S (or sometimes Ω, the Greek capital omega) and serves as the universal set for a given experiment. Every outcome of the experiment belongs to S, and probability is assigned to subsets of S called events. An event is simply a collection of outcomes that share some property we find interesting — for example, 'rolling an even number' or 'drawing a red card'. Because events are sets, we can combine them using union, intersection, and complement, giving probability theory a clean algebraic structure. Understanding sample spaces and events is not merely bookkeeping. The size and structure of the sample space determines which probability rules apply. A finite sample space with equally likely outcomes uses classical probability (counting methods), while continuous sample spaces require calculus-based techniques. High school probability overwhelmingly focuses on finite and discrete cases, but the vocabulary introduced here extends seamlessly into advanced statistics, quantum mechanics, and machine learning.

Key takeaways
  • Random Experiments
  • The Sample Space S
  • Events as Subsets
  • Set Operations on Events
  • Classical Probability Formula

Random Experiments

A random experiment is any process whose outcome cannot be determined with certainty before it is performed, but whose set of possible outcomes is known. Examples include tossing a coin, rolling a six-sided die, drawing a card from a deck, or selecting a student at random from a class. The key requirement is that the experiment can, in principle, be repeated under identical conditions.

The Sample Space S

The sample space S is the set of all possible outcomes of a random experiment. Each element of S is called an outcome or sample point. For a single coin toss: S = {H, T}. For a single die roll: S = {1, 2, 3, 4, 5, 6}. For two coin tosses: S = {HH, HT, TH, TT}, which has 2² = 4 elements. When rolling two dice, S has 6 × 6 = 36 ordered pairs (i, j) where i is the first die and j is the second. Writing out the full sample space prevents errors when computing probabilities.

Events as Subsets

An event E is any subset of the sample space: E ⊆ S. The event 'rolling an even number' on one die is E = {2, 4, 6}. The impossible event is the empty set ∅ (no outcomes), and the certain event is S itself. A simple event (or elementary event) contains exactly one outcome, e.g. {3}. A compound event contains two or more outcomes. The number of possible events for a sample space with n outcomes is 2ⁿ, since each outcome is either included or excluded.

Set Operations on Events

Because events are sets, three operations are fundamental. (1) Complement: The complement of E, written E' or Eᶜ, is the set of all outcomes in S not in E. If E = {2,4,6} and S = {1,2,3,4,5,6}, then E' = {1,3,5}. (2) Union: E₁ ∪ E₂ is the event that E₁ or E₂ (or both) occur — outcomes in at least one of the events. (3) Intersection: E₁ ∩ E₂ is the event that both E₁ and E₂ occur simultaneously. Two events are mutually exclusive (disjoint) if E₁ ∩ E₂ = ∅, meaning they cannot both happen at once.

Classical Probability Formula

When all outcomes in S are equally likely, the probability of event E is computed by counting: P(E) = |E| / |S|, where |E| denotes the number of outcomes in E. For example, the probability of rolling a prime number on a fair die: primes in {1,...,6} are {2,3,5}, so |E| = 3 and P(E) = 3/6 = 1/2. This formula only applies when outcomes are equally likely — never assume this without justification. Probabilities always satisfy 0 ≤ P(E) ≤ 1, P(∅) = 0, and P(S) = 1.

The Key Equation

P(E)=∣E∣∣S∣P(E) = \frac{|E|}{|S|}
  • P(E) — Probability of event E (dimensionless): A number between 0 and 1 representing how likely event E is to occur.
  • |E| — Cardinality of E (count): The number of outcomes in the event E (favourable outcomes).
  • |S| — Cardinality of S (count): The total number of outcomes in the sample space; must be finite and non-zero.

Worked Example

Problem. A standard six-sided die is rolled once. Define event A = 'the result is greater than 4' and event B = 'the result is even'. (a) List the sample space S. (b) List outcomes in A, B, A ∪ B, A ∩ B, and A'. (c) Find P(A), P(B), P(A ∪ B), and P(A ∩ B).

P(A)=13,P(B)=12,P(A∪B)=23,P(A∩B)=16P(A)=\tfrac{1}{3},\quad P(B)=\tfrac{1}{2},\quad P(A\cup B)=\tfrac{2}{3},\quad P(A\cap B)=\tfrac{1}{6}

Answer. P(A) = 1/3, P(B) = 1/2, P(A ∪ B) = 2/3, P(A ∩ B) = 1/6

Figure
Diagram
Sample Space S = {1, 2, 3, 4, 5, 6} for a Single Die RollSAB564213A = {5, 6} (greater than 4)B = {2, 4, 6} (even) A∩B = {6}
Origins · 1 minAndrey Kolmogorov and how this was discovered
AK
Andrey Kolmogorov
1903–1987 · Russian

Kolmogorov published his landmark work 'Foundations of the Theory of Probability' (Grundbegriffe der Wahrscheinlichkeitsrechnung) in 1933, establishing the rigorous axiomatic basis for modern probability theory. He defined sample spaces, events as measurable sets, and probability as a measure satisfying three axioms: non-negativity (P(E) ≥ 0), normalization (P(S) = 1), and countable additivity. This framework unified all earlier probability work and gave mathematics the precise language it needed for statistics, stochastic processes, and information theory. Kolmogorov made major contributions to at least seven different areas of mathematics, including topology, turbulence theory, and algorithmic complexity, and he was renowned as an exceptionally gifted teacher who personally mentored generations of Soviet mathematicians through hiking expeditions in the countryside.

  • Addition Rule of Probability — The addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) directly uses the union and intersection of events defined in the sample space framework.
  • Complementary Events — The complement rule P(E') = 1 − P(E) is a direct consequence of the sample space partition into E and its complement E'.
  • Conditional Probability — Conditional probability P(A|B) restricts the sample space to outcomes in B, making sample space reasoning essential for its calculation.
  • Set Theory and Set Operations — Events are subsets of the sample space, so union, intersection, and complement from set theory are the direct tools used throughout probability.