Sample space

In probability theory, a sample space—also called a sample description space, possibility space, or outcome space—is the set containing every possible result of an experiment or random trial. It is typically written in set notation, with each possible ordered outcome (called a sample point) listed as an element. Common labels for a sample space are S, Ω, or U (for "universal set"). The elements can be numbers, words, letters, or symbols, and the space itself may be finite, countably infinite, or uncountably infinite.

Any subset of the sample space is an event. If the outcome of an experiment falls inside that subset, the event is said to have occurred. For instance, tossing a single coin yields the sample space {H, T}, where H means heads and T means tails. The possible events are {H}, {T}, {H, T}, and the empty set. Tossing two coins gives the sample space {HH, HT, TH, TT}, where each pair describes the result of the first and second coin. The event that at least one coin is heads is {HH, HT, TH}. Rolling a single six-sided die once, with the number of pips facing up as the result, gives the sample space {1, 2, 3, 4, 5, 6}.

A well-defined, non-empty sample space is one of three components in a probabilistic model (a probability space). The other two are a well-defined set of possible events (an event space)—typically the power set of the sample space if it is discrete, or a σ-algebra if it is continuous—and a probability assigned to each event (a probability measure function). Visually, a sample space can be shown as a rectangle, with outcomes as points inside it. Events are represented by ovals that enclose the relevant points.

For a set S with outcomes sᵢ (so S = {s₁, s₂, ...}) to be a sample space, it must meet three conditions. First, outcomes must be mutually exclusive: if one outcome occurs, no other can. Second, outcomes must be collectively exhaustive: every experiment or random trial always produces some outcome in S. Third, the sample space must have the right granularity for what the experimenter cares about; irrelevant information should be removed, and the correct level of abstraction chosen. For example, in a coin toss, one possible sample space is {H, T}, while another could be {rainy day, non-rainy day}. For most experiments, {H, T} is the better choice, since weather does not affect the coin toss.

Many experiments have more than one plausible sample space, depending on what result interests the experimenter. Drawing a card from a standard 52-card deck, for instance, could have a sample space of ranks (Ace through King) or of suits (clubs, diamonds, hearts, spades). A more complete description would specify both denomination and suit, constructing the sample space as the Cartesian product of those two spaces—yielding 52 equally likely outcomes. Other sample spaces, such as right-side up or upside down, are also possible if some cards have been flipped during shuffling.

Some treatments of probability assume outcomes are always defined to be equally likely. For a sample space with N equally likely outcomes, each gets probability 1/N. But not every experiment fits this model: tossing a thumbtack many times and observing whether it lands point up or down has no physical symmetry to suggest the two outcomes are equally likely. Even so, defining a sample space so that outcomes are at least approximately equally likely can simplify probability calculations. If each outcome has the same probability, then the probability of any event becomes the number of outcomes in the event divided by the total number of outcomes. For example, throwing two fair six-sided dice to get two integers, X and Y, each from 1 to 6, yields 36 equally likely ordered pairs (X, Y). The probability that the sum is 5 is 4/36, because four of those pairs sum to five. If the sample space were instead all possible sums from rolling two dice, the same formula applies, but the number of outcomes in a given event varies: a sum of two occurs only with outcome (1,1), so its probability is 1/36; a sum of seven occurs with outcomes (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), so its probability is 6/36.