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In probability, a sample space is the foundation of every calculation and argument. It is the set you start with before you assign probabilities or talk about events.
A sample space (usually written $S$ or $\Omega$) is the set of all possible outcomes of a random experiment, with no outcome left out and no impossible outcome included.
An outcome is one possible result of the random experiment. The sample space is just “the collection of all those results.”
Basic examples of sample spaces
The form of a sample space depends on the experiment and on what you choose to record as an outcome.
- Tossing a single coin once
Experiment: Toss a fair coin and record which side is up.
A natural sample space is
$$
S = \{\text{Heads}, \text{Tails}\}.
$$
- Rolling a single die
Experiment: Roll a standard six-sided die and record the number of dots on the top face.
Sample space:
$$
S = \{1, 2, 3, 4, 5, 6\}.
$$
- Tossing two coins
Experiment: Toss two coins and record the result of each coin in order (first coin, second coin).
Sample space:
$$
S = \{\text{HH}, \text{HT}, \text{TH}, \text{TT}\}
$$
where, for example, HT means “first coin Heads, second coin Tails.”
Note that this is different from just writing $\{\text{H}, \text{T}\}$, because we are keeping track of both coins, not just whether at least one head appears.
- Rolling two dice
Experiment: Roll two distinguishable dice (for example, one red and one blue) and record the pair (red result, blue result).
A common sample space is the set of ordered pairs:
$$
S = \{(i,j) : i = 1,2,3,4,5,6;\ j = 1,2,3,4,5,6\}.
$$
This has $6 \times 6 = 36$ outcomes.
If you only cared about the sum of the two dice, you could instead choose
$$
S = \{2,3,4,5,6,7,8,9,10,11,12\},
$$
but notice that then each sum does not represent a single physical outcome; different rolls can give the same sum. This choice may or may not be convenient, depending on what you want to calculate.
- Drawing a card from a standard deck
Experiment: Draw one card from a standard 52-card deck.
One way to describe the sample space is:
$$
S = \{\text{all 52 distinct cards}\},
$$
for example including “Ace of spades,” “2 of hearts,” …, “King of clubs.” Each specific card is one outcome.
- Spinning a spinner
Suppose you have a circular spinner labeled with numbers from $0$ to $1$ continuously around the edge, and you record exactly where it lands.
A natural sample space is the interval
$$
S = [0,1],
$$
meaning any real number between $0$ and $1$, including the endpoints.
This is an example of an infinite sample space with continuously many outcomes.
Requirements for a sample space
When you define a sample space for an experiment, you must ensure:
- Every possible outcome is included
Nothing that can happen in the experiment should be missing from $S$. If your experiment is “roll one die,” then $7$ should not be in the sample space, because you cannot roll a $7$ on one standard die. But each of $1,2,3,4,5,6$ must be there.
- No impossible outcome is included
Including impossible outcomes can lead to confusion or incorrect probabilities.
- Outcomes are defined clearly
Each element of the sample space should have a clear meaning. For example, in tossing two coins, writing $S = \{\text{H}, \text{T}\}$ does not clearly distinguish between the outcomes of the first and second coin; it is therefore a poor choice if you need to describe, say, “first coin is heads, second is tails.”
- Exactly one outcome occurs on each trial
For each actual result of the experiment, there should be exactly one element of $S$ that describes it. This avoids ambiguity. If two different elements of $S$ could both describe the same physical result, your sample space is not well designed.
For instance, if you said
$$
S = \{\text{“even number”}, \text{“odd number”}, 1, 2, 3, 4, 5, 6\}
$$
for rolling a die, that would be a bad sample space: the outcome “4” and the outcome “even number” both describe the same result, so one roll could match more than one outcome in $S$.
Different sample spaces for the same experiment
The same physical experiment can have different sample spaces, depending on what information you want to keep track of.
- Two coin tosses
- If you care about the exact sequence, use:
$$
S_1 = \{\text{HH}, \text{HT}, \text{TH}, \text{TT}\}.
$$ - If you only care about the number of heads, you could instead use:
$$
S_2 = \{0, 1, 2\}
$$
where each number represents how many heads occur.
Both are valid sample spaces for the same experiment, just at different levels of detail. Later, the choice of sample space influences how you calculate probabilities.
- Rolling two dice
- Full detail (ordered pairs):
$$
S_1 = \{(i,j): i = 1,\dots,6;\ j = 1,\dots,6\}.
$$ - Only the sum:
$$
S_2 = \{2,3,4,5,6,7,8,9,10,11,12\}.
$$
The second sample space is smaller and may simplify some descriptions, but you lose the information about which individual faces appeared.
- Drawing a card
Depending on your question:
- If you care about the exact card:
$S = \{\text{all 52 individual cards}\}$. - If you only care about the suit, you might use:
$$
S = \{\text{hearts}, \text{diamonds}, \text{clubs}, \text{spades}\}.
$$ - If you only care whether it is a “face card” or “not a face card”, you could choose:
$$
S = \{\text{face}, \text{non-face}\}.
$$
Each is a valid sample space for a different question about the same physical experiment.
The key idea: the sample space should be detailed enough for the questions you want to answer, but not necessarily more detailed.
Finite and infinite sample spaces
Sample spaces can be:
- Finite
The sample space has a finite number of outcomes.
Examples:
- One coin: $S = \{\text{H}, \text{T}\}$ (2 outcomes).
- One die: $S = \{1,2,3,4,5,6\}$ (6 outcomes).
- Two dice (ordered pairs): 36 outcomes.
Many basic probability problems start with finite sample spaces.
- Countably infinite
The outcomes can be listed in a sequence (like $1, 2, 3, \dots$), but there are infinitely many.
Example experiment: Toss a coin repeatedly until the first heads appears, and record the number of tosses.
Sample space:
$$
S = \{1,2,3,4,5,\dots\}.
$$
You could in principle toss the coin many times before getting heads, so there is no fixed upper limit.
- Uncountably infinite (continuous)
Many experiments involving measurements have outcomes that can be any real number in some interval.
Examples:
- Measure the exact time (in seconds) that it takes for a computer to complete a task, rounded to infinite precision between $0$ and $10$ seconds:
$$
S = [0,10].
$$ - Spin a spinner labeled from $0$ to $1$:
$$
S = [0,1].
$$
Here there are infinitely many possible outcomes, and you cannot list them one by one; they fill an interval.
Sample points and events
Each individual element of the sample space is sometimes called a sample point (or an outcome).
An event is any subset of the sample space. The relationship is:
- Sample space $S$: “everything that can happen.”
- Sample point: a single element of $S$.
- Event: some collection of sample points.
For example, rolling a die with $S = \{1,2,3,4,5,6\}$:
- Sample points: $1$, $2$, $3$, $4$, $5$, $6$.
- Event “roll an even number”: $\{2,4,6\}$.
- Event “roll at least 5”: $\{5,6\}$.
Here, we are focused on the sample space itself; the ideas of events and operations on events are developed in more detail elsewhere.
Describing sample spaces clearly
When you write down a sample space, clarity matters. Common ways to describe sample spaces are:
- Listing all outcomes (roster form)
If there are only a few outcomes, simply list them, separated by commas and surrounded by braces.
- One die:
$$
S = \{1,2,3,4,5,6\}.
$$ - Two coin tosses:
$$
S = \{\text{HH},\ \text{HT},\ \text{TH},\ \text{TT}\}.
$$
- Using a rule or set-builder description
When there are many outcomes or a clear pattern, you might describe $S$ by a rule.
- Two dice:
$$
S = \{(i,j) : i=1,2,3,4,5,6;\ j=1,2,3,4,5,6\}.
$$ - Time between 0 and 10 seconds:
$$
S = \{t : 0 \le t \le 10\}.
$$
- Using intervals
For continuous outcomes, intervals of real numbers are common:
- $S = [0,1]$ for all real numbers from 0 to 1 inclusive.
- $S = (0,\infty)$ for all positive real numbers.
Whichever form you choose, the description should make it unambiguous which outcomes are in the sample space and which are not.
Common pitfalls in defining sample spaces
Several typical mistakes can cause trouble later when you assign probabilities:
- Leaving out possible outcomes
Example: For rolling a die, writing $S = \{1,2,3,4,5\}$ is missing the outcome 6. Any probability calculation based on this would be wrong.
- Combining outcomes that should be distinct
Example: For rolling two dice, writing $S = \{2,3,4,\dots,12\}$ if you intend this to represent each physical roll. The sum alone does not distinguish, for example, between $(1,6)$ and $(3,4)$; they are different physical outcomes with different properties in some contexts.
This is acceptable only if you really care only about the sums and not about which specific faces showed.
- Using vague descriptions
Example: For a card draw, writing $S = \{\text{“a card”}\}$ is not useful; it does not separate different possible cards. You need distinct outcomes that separate the different possibilities.
- Overlapping or redundant outcomes
Example: For rolling a die:
$$
S = \{\text{even number},\ \text{odd number},\ 1,\ 2,\ 3,\ 4,\ 5,\ 6\}
$$
is bad: the actual roll “4” fits both “even number” and “4”. A proper sample space must assign exactly one outcome to each physical result.
Careful thought at the stage of defining the sample space prevents many errors later.
Summary
- A sample space is the set of all possible outcomes of a random experiment.
- Each individual element of the sample space is an outcome (or sample point).
- The sample space must:
- Include all possible outcomes,
- Exclude impossible ones,
- Assign exactly one outcome to each possible result of the experiment.
- Different sample spaces can describe the same experiment at different levels of detail, depending on what you want to study.
- Sample spaces may be finite, countably infinite, or continuous (uncountably infinite).
- Clear and precise definition of the sample space is the first step in any probability problem.