Possibility Spaces
Sample Space (Outcome Space)
The sample space is the complete set of all possible outcomes of an experiment. Listing outcomes systematically avoids missing any.
Single Experiment
Example: Roll a fair six-sided die. List the sample space.
$$S = {1, 2, 3, 4, 5, 6}$$
$$P(\text{even}) = \frac{3}{6} = \frac{1}{2}$$
Combined Experiments — Possibility Spaces
When two (or more) experiments are combined, a two-way table (possibility space diagram) lists all outcomes.
Example: Two fair dice are rolled. The table shows all 36 equally likely outcomes (showing the sum):
| + | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
Total outcomes: $6 \times 6 = 36$
$$P(\text{sum} = 7) = \frac{6}{36} = \frac{1}{6}$$ $$P(\text{sum} > 9) = \frac{6}{36} = \frac{1}{6}$$
Spinner and Coin Combined
Example: A spinner has sections 1, 2, 3. A coin is flipped. List all outcomes.
| H | T | |
|---|---|---|
| 1 | 1H | 1T |
| 2 | 2H | 2T |
| 3 | 3H | 3T |
Total: $3 \times 2 = 6$ equally likely outcomes.
$$P(\text{even number and Heads}) = \frac{1}{6}$$
Product Rule for Counting
If event A has $m$ equally likely outcomes and event B has $n$ equally likely outcomes, then the combined experiment has $m \times n$ outcomes.
This is why a possibility space for two dice has $6 \times 6 = 36$ cells.
Using Possibility Spaces to Find Probabilities
- Construct the full table (show all cells — marks are awarded for a complete table).
- Count the favourable outcomes.
- Divide by the total number of outcomes.
$$P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$$
WJEC Exam Tips
- Always draw the complete possibility space table — partial tables lose marks.
- Label rows and columns clearly.
- Two dice problems: sum, difference, product — know which table to build.
- For the WJEC Intermediate/Higher tier: conditional probability can be read from a possibility space table.
AI-generated · claude-opus-4-7 · v3-wjec-maths