Sample Spaces and Events

Choose the level of detail

For one fair die, $S=\{1,2,3,4,5,6\}$ is a natural sample space. For two distinguishable fair dice rolled independently, use ordered pairs:

There are $6\times6=36$ ordered pairs, each with probability $1/36$. The pair $(2,5)$ differs from $(5,2)$ because it identifies which die showed which result. The labels could equally represent a first and a second roll of the same die.

If the only recorded quantity is the sum, the possible values are $2,3,\ldots,12$. That is a valid reduced description, but it has eleven unequally likely values. A sum of two comes from one ordered pair; a sum of seven comes from six. Reduced spaces must retain those different weights.

List an event before calculating

Let $A$ be the event that the sum is seven. Then

so $P(A)=6/36=1/6$. Let $B$ be the event of doubles. It consists of $(1,1),(2,2),\ldots,(6,6)$, also giving $P(B)=1/6$.

These two events have the same probability but different members. Their intersection is empty because a double has an even sum. Their union therefore contains twelve outcomes and has probability $12/36=1/3$.

Equal probabilities do not make events identical. Nor do they establish independence. Here the events cannot both occur, so they are dependent because both have positive probability.

Translate event language into set operations

For one fair die, let $E=\{2,4,6\}$ be the even outcomes and $G=\{4,5,6\}$ the outcomes greater than three. Then

The intersection $E\cap G$ means both events occur. The union $E\cup G$ means at least one occurs, including both. The complement $E^c$ contains the outcomes in $S$ that are not in $E$, so $E^c=\{1,3,5\}$.

“Exactly one of E and G” excludes their overlap. Its outcomes are $\{2,5\}$, not the entire union. “Neither” means $(E\cup G)^c=\{1,3\}$.

The complement always depends on the stated sample space. If the experiment changes to a ten-sided die, the same words may define different sets and probabilities.

Use a partition to prevent double-counting

A partition divides the sample space into events that do not overlap and together cover every outcome. For the two-dice experiment, the events “sum less than seven”, “sum equals seven” and “sum greater than seven” form a partition. They contain 15, 6 and 15 outcomes, respectively.

The events “sum at most seven” and “sum at least seven” do not form a partition: both contain the sum-seven outcomes. A total greater than one would be a warning that outcomes had been counted more than once, not evidence that probability itself exceeds one.

Label objects when repeated categories hide outcomes

Suppose a bag contains A1, A2, A3, B1 and B2. Drawing two distinct tokens without replacement gives $5\times4=20$ ordered elementary outcomes under equal selection at each step. The category sequence AB represents six of them: any of three A tokens followed by either B token.

The four category sequences AA, AB, BA and BB have elementary counts 6, 6, 6 and 2. Their probabilities are $3/10$, $3/10$, $3/10$ and $1/10$. Listing four category sequences and assigning each probability $1/4$ would lose the original selection model.

An unordered two-token selection is another valid representation. Its ten subsets are equally likely under the same selection procedure, and six contain one token of each type. Both ordered and unordered methods give $6/10=3/5$ for one A and one B when their numerators and denominators use the same kind of outcome.

Some sample spaces are not finite lists

If independent fair tosses continue until the first head, the stopping outcomes are $H,TH,TTH,\ldots$. Their probabilities are $1/2,1/4,1/8,\ldots$, which sum to one. In the full infinite-sequence model, the endless-tail sequence also exists but has probability zero. The stopping-count distribution describes the finite outcomes that occur with total probability one.

A measurement model, such as a time anywhere in an interval, instead uses a continuous space. The elementary counting formula $|A|/|S|$ is not the method for such a space. Probability Distributions introduces interval areas and densities.

Related topics

Complementary Events develops “not” and “at least” reasoning. Counting Principle provides efficient alternatives when a complete list becomes too large.