Probability
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.
Understanding the sheer size of a sample space explains why some events feel almost impossible: a typical national lottery draw, choosing 6 numbers from 49, has a sample space of nearly 14 million equally likely combinations, which is exactly why the odds of matching all six are roughly 1 in 14 million. Card games reach even more staggering scales – the sample space for shuffling a standard 52-card deck contains about 8 × 1067 possible orderings, a number so large that every time a deck is properly shuffled, the exact resulting order has almost certainly never existed before in the history of card games.
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
S = {HH, HT, TH, TT}. Total outcomes = 4.
| 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.
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.
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.