Choosing the Sample Space

Why the Choice of Sample Space Matters

A sample space is a description of the possible outcomes of an experiment. The description can preserve many details, such as the order of results or the identity of each object, or it can summarize them. Both choices may correctly describe what was recorded, but they do not necessarily answer the same questions. A sample space is useful when it keeps the distinctions needed for the questions being asked.

For example, recording only the sum of two die rolls is enough to answer whether the sum is 7. It does not reveal which number appeared on the first roll. To answer that question, the sample space must preserve the two results separately. The central issue is not whether one description is always best. It is whether the outcomes in the description retain the information relevant to the task.

Definition: A sample space is a collection of possible outcomes for an experiment, with each outcome description representing a distinct result at the chosen level of detail. A coarser sample space records less detail by grouping results that a more detailed sample space distinguishes. A question is answerable from a sample space if each outcome description gives enough information to determine its answer.

In the finite examples here, an event is a yes-or-no question about the result. We can describe its “yes” outcomes as a subset of the sample space. If two detailed outcomes receive the same simplified description, then any question answerable from the simplified description must give the same answer for both. That observation gives a precise test for whether a sample space has enough detail.

A Criterion for Whether a Question Can Be Answered

Suppose $\Omega$ is a detailed sample space and $\Gamma$ is a simpler one. A function $g:\Omega\to\Gamma$ describes how each detailed outcome is recorded in the simpler model. The detailed outcomes that lead to a particular recorded outcome are grouped together by $g$. For example, if $g$ reports the sum of two die results, all detailed pairs with the same sum are grouped together.

Let $A\subseteq\Omega$ be the detailed outcomes for which a question has answer “yes.” The question can be answered from the simplified outcome $g(\omega)$ precisely when no two detailed outcomes grouped together by $g$ give different answers.

Theorem (Representability criterion): Let $\Omega$ and $\Gamma$ be finite sample spaces, let $g:\Omega\to\Gamma$, and let $A\subseteq\Omega$ describe the “yes” outcomes for a question. The question can be answered from the simplified outcome $g(\omega)$ if and only if $$ g(\omega)=g(\omega') \quad\Longrightarrow\quad \bigl(\omega\in A \ \text{if and only if}\ \omega'\in A\bigr) $$ for all $\omega,\omega'\in\Omega$.

Proof. If the question can be answered from the simplified outcome, there is a rule that assigns “yes” or “no” to each value in $\Gamma$. If $g(\omega)=g(\omega')$, the rule receives the same input for both outcomes and must give the same answer. Thus $\omega$ and $\omega'$ either both belong to $A$ or both lie outside $A$.

Conversely, suppose the displayed condition holds. For any $y\in\Gamma$ that is the simplified description of at least one outcome, assign “yes” if an outcome $\omega$ with $g(\omega)=y$ belongs to $A$, and assign “no” otherwise. This assignment is well-defined: the condition ensures that all detailed outcomes with simplified description $y$ agree on the answer. Any $y$ that is not attained by $g$ can be assigned either answer, since it cannot occur. The question is therefore answerable from the simplified outcome. $\square$

Worked Example: Two Dice and Their Sum

Roll a red die and then a blue die. The detailed sample space records the ordered pair $(r,b)$, where each entry is in $\{1,2,3,4,5,6\}$. Consider the simplified description $g(r,b)=r+b$, which records only the sum.

The question “Is the sum 7?” is answerable from the simplified outcome: the answer is yes exactly when the recorded sum is 7. Every detailed pair with a given sum has the same answer to this question.

Now consider “Did the red die show 1?” The detailed outcomes $(1,2)$ and $(2,1)$ both have simplified description 3. The answer is yes for $(1,2)$ and no for $(2,1)$. The representability criterion fails, so the sum alone cannot answer the question.

The ordered-pair sample space preserves the identity of each die’s result. The sum sample space discards that distinction while retaining exactly the information needed for questions about the total. Which is appropriate depends on the question.

Worked Applications at Different Levels of Detail

Worked Example: Drawing Two Cards

Draw two cards successively from a standard 52-card deck, without replacing the first card. One detailed description records the exact ordered pair of cards. There are $52\cdot 51=2652$ possible ordered pairs: 52 choices for the first card and then 51 remaining choices for the second.

A simpler description records only the two ranks, without order. This description has enough information to answer “Are both cards kings?” The answer is yes exactly when both recorded ranks are king. It also answers “Did the two cards have the same rank?”

However, the rank-only description cannot answer “Was the first card the ace of spades?” For example, the ordered draws (ace of spades, 7 of hearts) and (7 of hearts, ace of spades) have the same unordered rank description, but only the first has the ace of spades first. The exact ordered-card sample space can answer the question; the simplified description cannot.

A different simplification records only whether the two ranks match. That is enough for the question “Did the ranks match?” but loses even more information: it cannot determine whether either card was a king. Each simplification supports some questions and rules out others.

Worked Example: Testing Four Components

A technician tests four identified components, labeled A, B, C, and D, and records whether each passes or fails. A detailed outcome is a four-entry sequence, such as (pass, fail, pass, pass). There are $2^4=16$ possible sequences because each of the four components has two possible test results.

Suppose a simplified description records only the number of failed components. Its possible values are 0, 1, 2, 3, and 4. It answers “Did exactly two components fail?” because that is determined by the count.

The count cannot answer “Did component A fail?” For example, (fail, pass, pass, pass) and (pass, fail, pass, pass) both have one failure, but only the first has a failure at A. The detailed sequence distinguishes these outcomes; the failure count groups them together.

The count is a suitable sample space if the questions concern only how many components failed. If the technician needs to identify which components failed, the sequence is needed. Recording only the count is not inherently incorrect; it is insufficient for questions that depend on component identity.

Comparing Sample Spaces

The examples show a general direction: a detailed description can often be simplified by forgetting information, but simplifying may make some questions unanswerable. Formally, if $g:\Omega\to\Gamma$ converts detailed outcomes into simplified outcomes, any question that can be answered from $\Gamma$ can also be answered from $\Omega$: given $\omega$, compute $g(\omega)$ and then use the answer rule for $\Gamma$.

Theorem (A coarser description cannot add detail): Let $g:\Omega\to\Gamma$ map a detailed outcome to its simplified description. Every question answerable from $\Gamma$ is answerable from $\Omega$. The reverse need not hold.

Proof. If a question is answerable from $\Gamma$, there is a rule $h$ that determines its answer from the simplified outcome. For a detailed outcome $\omega$, first calculate $g(\omega)$, then apply $h$. The rule $h(g(\omega))$ gives the answer using only $\omega$, so the question is answerable from $\Omega$. The reverse need not hold: in the dice example, “Did the red die show 1?” is answerable from the ordered pair but not from the sum. $\square$

This comparison is about what can be determined, not about which description is more probable or more appropriate for every calculation. A detailed space may be convenient for tracking distinct outcomes; a simpler one may be easier to work with for a particular question. Before choosing, identify what information the question requires and check whether the proposed outcomes preserve it.

Choosing Carefully: Common Pitfalls

Keeping too little detail. If a sample space records only a sum, it cannot answer questions about which die showed which value. Check whether outcomes grouped together could produce different answers.
Keeping more detail than the question needs. A detailed list of card identities is valid for asking whether two ranks match, but that question can also be answered from a simpler rank-based description. Extra detail is not a mistake, though it may make the model less convenient.
Confusing a summary with the experiment itself. “Number of failures” is a description derived from the component test results. It does not say which components failed, so it should not be used to answer identity-specific questions.
Assuming that outcomes in a chosen space are equally likely. The way outcomes are recorded does not, by itself, assign probabilities. For example, sums from two dice do not all arise from the same number of ordered pairs: sum 2 arises from one pair, while sum 7 arises from six. A probability model must reflect the experiment, not just the number of labels in a summary.

A practical choice is to begin with the distinctions the experiment genuinely records, then simplify only when the questions of interest do not depend on the discarded information. This makes the sample space a deliberate modeling decision rather than a list chosen by habit.

Check Your Understanding

For each question, decide what information a suitable sample space must preserve.

Two dice are rolled and only their sum is recorded. Can this description answer whether the first die showed 4? Explain using two outcomes with the same sum.
Two cards are drawn in order. Would recording only the unordered pair of ranks answer whether the first card was a heart? Why or why not?
Five components are tested. Compare the information in a five-entry pass/fail sequence with the number of failures.
State the representability criterion in words: when can two detailed outcomes be grouped into one simplified outcome without losing the answer to a particular question?
Why does a sample space with fewer outcome labels not automatically imply that all its outcomes have the same probability?

What You'll Learn