Sample Space

Q: How do you find the sample space when there are multiple steps in an experiment?

Use the counting principle: if step 1 has $m$ outcomes and step 2 has $n$ outcomes, the sample space has $m \times n$ total outcomes. You can also draw a tree diagram to systematically list every combination. For instance, flipping a coin and then rolling a die produces $2 \times 6 = 12$ outcomes.

Sample Space

The set of all possible outcomes of an experiment.

Example 1:

A coin is flipped.

Sample Space = {heads, tails}

Example 2:

Two six-sided dice are rolled.

If (3, 5) means the first die shows 3 and the second die shows 5, then

Sample Space = A 6×6 grid showing all ordered pairs (1,1) through (6,6) representing the sample space for rolling two six-sided dice.

See also

Event, probability

Key Formula

P(E) = \frac{\text{Number of outcomes in } E}{\text{Number of outcomes in } S}

Where:

$P(E)$ = Probability of an event E
$E$ = An event, which is a subset of the sample space
$S$ = The sample space (set of all possible outcomes)

Worked Example

Problem: A standard six-sided die is rolled once. List the sample space, then find the probability of rolling an even number.

Step 1: Identify every possible outcome when rolling one die. Each face shows a number from 1 to 6.

S = \{1, 2, 3, 4, 5, 6\}

Step 2: Count the total number of outcomes in the sample space.

|S| = 6

Step 3: Define the event E as rolling an even number, and list its outcomes.

E = \{2, 4, 6\}

Step 4: Calculate the probability by dividing the number of outcomes in E by the total number of outcomes in S.

P(E) = \frac{|E|}{|S|} = \frac{3}{6} = \frac{1}{2}

Answer: The sample space is

S = \{1, 2, 3, 4, 5, 6\}

, and the probability of rolling an even number is

\frac{1}{2}

Another Example

Problem: A coin is flipped twice. List the complete sample space and find the probability of getting exactly one head.

Step 1: For each flip, there are two outcomes: Heads (H) or Tails (T). Since the coin is flipped twice, list every ordered pair of results.

S = \{HH, HT, TH, TT\}

Step 2: Count the total outcomes. Two flips with 2 outcomes each gives

2 \times 2 = 4

outcomes.

|S| = 4

Step 3: Define event E as getting exactly one head. Pick out the outcomes with one H and one T.

E = \{HT, TH\}

Step 4: Calculate the probability.

P(E) = \frac{2}{4} = \frac{1}{2}

Answer: The sample space has 4 outcomes:

\{HH, HT, TH, TT\}

. The probability of exactly one head is

\frac{1}{2}

Frequently Asked Questions

What is the difference between a sample space and an event?

The sample space is the complete set of every possible outcome of an experiment. An event is any subset of the sample space — it contains one or more specific outcomes you care about. For example, when rolling a die, the sample space is

\{1,2,3,4,5,6\}

, while the event 'rolling greater than 4' is the subset

\{5,6\}

How do you find the sample space when there are multiple steps in an experiment?

Use the counting principle: if step 1 has

m

outcomes and step 2 has

n

outcomes, the sample space has

m \times n

total outcomes. You can also draw a tree diagram to systematically list every combination. For instance, flipping a coin and then rolling a die produces

2 \times 6 = 12

outcomes.

Sample Space vs. Event

	Sample Space	Event
Definition	The set of all possible outcomes of an experiment	Any subset of the sample space
Size	Contains every possible outcome	Can contain zero, one, some, or all outcomes
Example (rolling a die)	$S = \{1,2,3,4,5,6\}$	'Rolling odd' = $\{1,3,5\}$
Relationship	The universal set for the experiment	Always a subset of the sample space

Why It Matters

The sample space is the starting point for all probability calculations. Without correctly identifying every possible outcome, you cannot accurately determine how likely any event is. In more advanced topics like statistics and data science, properly defining the sample space ensures that probability models reflect reality.

Common Mistakes

Mistake: Forgetting to count order when it matters. For example, when rolling two dice, treating (3, 5) and (5, 3) as the same outcome.

Correction: If the dice are distinguishable (e.g., one red, one blue), (3, 5) and (5, 3) are different outcomes. Rolling two dice produces

6 \times 6 = 36

outcomes, not 21. Always ask whether order matters before listing the sample space.

Mistake: Listing outcomes that are not equally likely and then using the simple probability formula.

Correction: The formula

P(E) = \frac{|E|}{|S|}

only works when every outcome in

S

is equally likely. If outcomes have different probabilities — like drawing from a weighted spinner — you must assign individual probabilities to each outcome instead.

Related Terms

Set — A sample space is a specific type of set
Outcome — Each element of a sample space
Experiment — The process that generates the sample space
Event — A subset of the sample space
Probability — Calculated using the sample space
Counting Principle — Used to find the size of a sample space
Tree Diagram — Visual tool for listing a sample space