Possibility Spaces
Possibility Spaces
A possibility space (or sample space diagram) is a systematic grid showing all possible outcomes of a two-stage experiment. It is particularly useful when both stages involve the same type of variable (e.g. two dice) and allows probabilities to be read off by counting cells.
Constructing a Possibility Space
- List the outcomes of the first stage along one axis and the second stage along the other.
- Fill each cell with the combined outcome (e.g. sum, product, or pair).
- Count favourable cells and divide by the total number of cells to find the probability.
Example — Sum of Two Dice
Rolling two fair dice: 36 equally likely outcomes. The possibility space for 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 |
\( P(\text{sum}=7)=6/36=1/6 \). \( P(\text{sum}\geq10)=6/36=1/6 \). \( P(\text{sum is prime})=15/36=5/12 \).
Using a Possibility Space
Once the grid is completed, circle or shade all cells satisfying the event and count them. Divide by the total (which equals the product of the number of outcomes on each axis).
Key Takeaways
- Possibility space: grid of all equally likely combined outcomes; total = (outcomes stage 1) × (outcomes stage 2).
- Probability = number of favourable cells ÷ total cells.
- All cells are equally likely only when both stages have equally likely outcomes.
- The most common sum from two standard dice is 7 (6 ways out of 36).
- Possibility spaces work best for two-stage experiments; use tree diagrams for three or more stages.