Sample Space - Every Possible Outcome
The sample space is the complete set of all possible outcomes of a probability experiment. Knowing the sample space fully and accurately is essential before you can calculate any probability. Miss an outcome and your probability calculations will be wrong.
Notation
The sample space is usually denoted by the letter S (or sometimes Ω, the Greek capital omega). Its outcomes are listed inside curly braces.
Example: rolling a six-sided die → S = {1, 2, 3, 4, 5, 6}.
Methods for Finding the Sample Space
| Method | Best used when |
|---|---|
| List (roster notation) | Small number of outcomes that are easy to name |
| Two-way table (grid) | Two experiments combined (e.g. two dice) |
| Tree diagram | Multi-step experiments where order matters |
Worked Examples
Write the sample space for tossing two coins.
S = {HH, HT, TH, TT}. Total outcomes = 4.
Two dice are rolled. Write the sample space using a grid and find the total number of outcomes.
| Die 1 \ Die 2 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | (1,1) | (1,2) | (1,3) | (1,4) | (1,5) | (1,6) |
| 2 | (2,1) | (2,2) | (2,3) | (2,4) | (2,5) | (2,6) |
| 3 | (3,1) | (3,2) | (3,3) | (3,4) | (3,5) | (3,6) |
| 4 | (4,1) | (4,2) | (4,3) | (4,4) | (4,5) | (4,6) |
| 5 | (5,1) | (5,2) | (5,3) | (5,4) | (5,5) | (5,6) |
| 6 | (6,1) | (6,2) | (6,3) | (6,4) | (6,5) | (6,6) |
Total outcomes = 6 × 6 = 36.
Using the two-dice grid above, find P(sum = 7).
Pairs that sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) – that is 6 outcomes.
P(sum = 7) = 6/36 = 1/6.
A letter is chosen at random from the word MATH. Write the sample space and find P(vowel).
S = {M, A, T, H}. Vowels = {A}. P(vowel) = 1/4 = 0.25.
Key Takeaways
- The sample space S is the complete list of all possible outcomes of an experiment.
- Use a list for simple experiments, a table for two combined experiments, and a tree diagram for multi-step experiments.
- P(event) = outcomes in the event ÷ total outcomes in S.
- Always check your sample space is complete – a missing outcome changes all probabilities.
Practice Questions
- Write the sample space for rolling a four-sided die (faces 1 to 4) and tossing a coin.
- Draw a grid for rolling two dice and use it to find P(sum = 9).
- A spinner has sections A, B, C. It is spun twice. Write the full sample space.
- Using the two-dice grid, find P(both numbers the same).
- A bag has cards labelled 1, 2, 3, 4, 5. One card is drawn. Write S and find P(prime number).