Sample Space

A sample space is the set of all possible outcomes of a random experiment, as represented by a probability model. Usually denoted by $\Omega$, $S$, or $\mathcal S$, it specifies what can happen, rather than how likely each outcome is. Its elements, called outcomes or sample points, may be numbers, labels, ordered sequences, or more complex objects. Together with a collection of events and a probability measure, the sample space forms a probability space. (ocw.mit.edu)

Outcomes and events

A sample space describes outcomes at a chosen level of detail. Its outcome descriptions must be mutually exclusive and collectively exhaustive: one trial produces exactly one represented outcome, and every result admitted by the model has a representation. For a coin toss recording the upper face, for example, $\Omega=\{H,T\}$. A model that also records whether the coin lands on its edge would require an additional outcome. (live.ocw.mit.edu)

An event is a subset of the sample space to which the model assigns a probability. It occurs when the observed sample point belongs to that subset. For a six-sided die,

and the event “an even number appears” is $A=\{2,4,6\}$. The whole space represents the certain event; the empty set represents the impossible event. (ocw.mit.edu)

The operations of set theory express relationships between events. The union $A\cup B$ means that at least one occurs, the intersection $A\cap B$ means that both occur, and the complement $\Omega\setminus A$ means that $A$ does not occur. These operations distinguish an individual outcome from a statement that several different outcomes could satisfy. (ocw.mit.edu)

Finite and countably infinite spaces

A finite sample space can be listed explicitly. Two coin tosses, with order retained, give

More generally, a sequence of $n$ coin tosses has $2^n$ possible face sequences. This construction records the full sequence rather than only the number of heads. (math.cmu.edu)

In the usual model on a finite or countably infinite set, every subset is an event, so the event collection is the power set $\mathcal P(\Omega)$. A probability mass function assigns nonnegative weights $p(\omega)$ satisfying

The set itself does not determine these weights. (math.cmu.edu)

For a finite space with equally likely outcomes, probabilities reduce to counting:

where vertical bars denote cardinality. Equal likelihood is an additional modeling assumption, not a consequence of listing the outcomes. Changing a coin from fair to biased changes its probability law without changing the two-element sample space. Counting methods from combinatorics are therefore applicable only after the relevant weights have been established. (math.mit.edu)

Continuous spaces and measurable events

An uncountable sample space may represent a continuously varying result. Selecting a real number uniformly from the interval $[0,1]$, for example, uses $\Omega=[0,1]$. Under the uniform distribution, an interval $[a,b]$ has probability $b-a$, while each individual point has probability zero. Thus, membership in the sample space does not imply positive probability. (math.cmu.edu)

In the framework of measure theory, the model is a triple

Here $\mathcal F$ is a sigma-algebra of subsets of $\Omega$, containing the whole space and closed under complements and countable unions. The probability measure is defined on $\mathcal F$, not necessarily on every subset of $\Omega$. It satisfies $P(\Omega)=1$, nonnegativity, and countable additivity for disjoint events. (math.mit.edu)

For real-valued models, a standard choice is the Borel sigma-algebra, generated by open sets. The uniform model on $[0,1]$ can use the restriction of Lebesgue measure to this interval. Restricting the event collection avoids nonmeasurable subsets that prevent assigning probabilities consistently with the intended geometric interpretation. (math.mit.edu)

Random variables and alternative representations

A random variable is a measurable function defined on the sample space. For a real-valued variable,

An outcome $\omega$ is the underlying result; $X(\omega)$ is the numerical value extracted from it. Different outcomes can produce the same value. The resulting probability distribution is determined by

for Borel sets $B$. (math.mit.edu)

For two distinguishable dice, a detailed sample space is the Cartesian product

The sum variable $X(i,j)=i+j$ takes values in $\{2,\ldots,12\}$, which can instead serve as the sample space of a model recording only the sum. These eleven sums are not equally likely under independent fair rolls: one ordered pair produces 2, whereas six produce 7. A coarser representation must retain the probabilities inherited from the detailed model. (math.dartmouth.edu)

Compound experiments

Product spaces represent experiments with several components. Their probability laws need not make those components independent: statistical independence concerns the probability measure, rather than merely the product structure of the underlying set. This separates the description of possible combinations from assumptions about their likelihoods. (math.dartmouth.edu)

Infinite sequences require a further distinction. For infinitely many coin tosses, $\Omega=\{H,T\}^{\mathbb N}$ contains complete sequences, although an event may specify only finitely many coordinates. Under independent fair tossing, the event fixing the first $n$ faces has probability $2^{-n}$. An appropriate sigma-algebra supports these probabilities even though the full power set cannot support the intended law. (math.cmu.edu)