Sample Spaces — Listing Outcomes for a Single Event
A sample space is the complete list of all possible outcomes for an experiment. Every outcome must appear exactly once. Once you have the sample space, use the probability formula to find any probability.
Example 1 — Single die: writing the sample space and finding a probability
Step 1
Write the sample space: S = {1, 2, 3, 4, 5, 6} — 6 equally likely outcomes.
Step 2
Find P(number less than 4). Favourable outcomes: {1, 2, 3} — 3 outcomes.
Step 3
P(less than 4) = \(\frac{3}{6} = \frac{1}{2}\)
Example 2 — Spinner with repeated sections
Scenario
A spinner has 8 equal sections: 3 Red, 3 Blue, 2 Green. Find P(Red) and P(not Red).
Mutually exclusive outcomes cannot happen at the same time — e.g. rolling a 3 and a 5 on one throw. Exhaustive outcomes cover all possibilities, so their probabilities sum to 1.
✓ Quick Check 1 — Sample Spaces and Single Events
Question 1 of 10
Listing Outcomes for Two Events — Sample Space Diagrams
For two independent events, list all combinations systematically. The total number of outcomes = (outcomes for event 1) × (outcomes for event 2). A sample space diagram (grid) arranges these in a table for easy reading.
Total combinations = outcomes of Event 1 × outcomes of Event 2
Sample Space Grid — Two Dice (36 outcomes). Gold cells: sum = 7
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Example 3 — Two dice: using the grid to find P(sum = 7)
Step 1
Total outcomes = 6 × 6 = 36
Step 2
From the grid, cells showing sum 7 (highlighted in gold): (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 cells
Step 3
P(sum = 7) = \(\frac{6}{36} = \frac{1}{6}\) — the most likely total when rolling two dice.
Example 4 — Two dice: P(sum = 10)
Step 1
Locate cells in the grid where the sum is 10: (4,6), (5,5), (6,4) = 3 cells
Step 2
P(sum = 10) = \(\frac{3}{36} = \frac{1}{12}\)
Be systematic: Fix all outcomes for one event and vary the other. This prevents missing combinations or double-counting.
Another way to list outcomes for two events is a tree diagram — you'll cover this fully in Lesson 14.5 — Tree Diagrams.
✓ Quick Check 2 — Two Events and Sample Space Diagrams
Question 1 of 10
Practice: Listing Outcomes
Enter your answers as numbers.
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Question
Answer
Result
1
How many outcomes are there when flipping 2 coins?
2
How many outcomes are there when rolling a die and flipping a coin?
3
A meal deal has 3 sandwich choices and 2 drink choices. How many different combinations are possible?
4
How many total outcomes are there when rolling two dice?
5
A padlock code uses 2 digits, each from 0–9. How many different codes are possible?
6
Two fair dice are rolled (36 total outcomes). P(sum = 2)?
7
Two fair dice are rolled. P(both dice show the same number)?
8
Two fair dice are rolled. P(sum > 10)?
9
Two fair dice are rolled. P(at least one 6)?
10
Two fair dice are rolled. P(sum is even)?
⚠ Things to Watch Out For
Missing outcomes in a systematic list: Work through outcomes in a fixed order (e.g. fix the first variable, vary the second). Random listing almost always misses some outcomes.
Duplicating outcomes: (H, T) and (T, H) are DIFFERENT outcomes — they describe different results. Don't treat them as the same.
Sample space size: For two independent events with m and n outcomes each, total outcomes = m × n. Two dice: 6 × 6 = 36 outcomes total.
Reading the question carefully: "At least one head" includes (H,H), (H,T), and (T,H). Listing only (H,T) misses outcomes. "At least one" means one or more.
Sample space grid: Use a grid for two combined events. Label rows with one event's outcomes and columns with the other's. Each cell is one possible combined outcome.