Listing Outcomes and Possibility Spaces
Whether you are choosing a pizza topping, predicting a football score, or designing a fair game — you are using probability. The secret to getting probabilities right is listing every possible outcome systematically. This guide shows you exactly how.
Key Definitions
Outcome
One possible result of an experiment. Rolling a die gives outcomes 1–6. Flipping a coin gives Heads or Tails.
Possibility Space
The complete set of all possible outcomes — also called the sample space. For a coin: {H, T}. For a die: {1,2,3,4,5,6}.
Probability Formula
An event is the specific outcome or group of outcomes you are interested in.
Three Ways to List Outcomes
1. Systematic Lists
Best for one event or simple sequential experiments.
Example: Flip a coin twice. Fix the first flip, then list all options for the second:
- First H: HH, HT
- First T: TH, TT
Possibility space: {HH, HT, TH, TT}
2. Two-Way Tables (Grids)
Ideal for two combined experiments. One experiment heads the columns, the other heads the rows.
Example: Flip a coin and roll a die:
| H | T | |
|---|---|---|
| 1 | (H,1) | (T,1) |
| 2 | (H,2) | (T,2) |
| 3 | (H,3) | (T,3) |
| 4 | (H,4) | (T,4) |
| 5 | (H,5) | (T,5) |
| 6 | (H,6) | (T,6) |
Total: $2 \times 6 = 12$ outcomes.
3. Venn Diagrams
Use when elements can belong to overlapping groups — for example, students who play two sports.
How to fill in a Venn diagram:
- Overlap first: 3 play both → middle.
- Only F: $12 – 3 = 9$ → left.
- Only R: $8 – 3 = 5$ → right.
- Neither: $20 – 17 = 3$ → outside.
Worked Examples
Example 1: Two Dice
Two dice are rolled and added. Find: (a) $P(\text{sum}=7)$ (b) $P(\text{sum}<5)$.
| + | 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: 36 outcomes.
(a) 6 sums of 7 (highlighted): $P = \frac{6}{36} = \mathbf{\frac{1}{6}}$.
(b) Sums 2,3,4: $1+2+3=6$ outcomes: $P = \frac{6}{36} = \mathbf{\frac{1}{6}}$.
Example 2: Choosing an Outfit
3 T-shirts (R, B, G) and 2 trousers (J, C). What is $P(\text{Blue and Jeans})$?
List systematically by T-shirt:
- (R,J), (R,C)
- (B,J), (B,C)
- (G,J), (G,C)
Total: $3 \times 2 = 6$ outcomes. One favourable.
$P(\text{Blue and Jeans}) = \mathbf{\dfrac{1}{6}}$.
Tutor Insights
🤔 Common Misunderstandings
- Not being systematic. Randomly listing outcomes leads to missing one or two — instantly making all probabilities wrong.
- Confusing outcomes and events. An outcome is a single result (e.g., (H,6)); an event is a group fitting a description (e.g., “heads and even”).
- Forgetting to simplify fractions in the final answer.
📝 Common Exam Mistakes
- Incomplete possibility space — a missing outcome makes every subsequent probability wrong.
- Arithmetic slips when computing sums in a two-dice table.
- Miscounting favourable outcomes — go through the grid methodically, not by eye.
- Writing probability > 1 — probability must be between 0 and 1.
Practice Questions
- A bag contains letters M, A, T, H, S. One is picked at random.
(a) Write the possibility space. (b) Find $P(\text{picking A})$. - A spinner {1,2,3,4} and a coin are used together.
(a) Draw a two-way table. (b) How many outcomes? (c) Find $P(\text{odd number and Tail})$. (d) Find $P(\text{number} > 2)$. - Spinner A {1,2,3}, Spinner B {4,5}. Scores multiplied.
(a) Construct the possibility space table. (b) Find $P(\text{product}=10)$. (c) Find $P(\text{product is even})$. - 50 students: 30 like Salt & Vinegar, 25 like Cheese & Onion, 10 like both.
(a) Draw a Venn diagram. (b) How many like only S&V? (c) How many like neither? (d) Find $P(\text{only C\&O})$. - Menu: 3 starters, 2 mains, 2 desserts.
(a) List all three-course combinations. (b) How many total? (c) Find $P(\text{Soup and Cake})$.
Show Answers
- (a) {M,A,T,H,S}. (b) $\frac{1}{5}$.
- (b) $4\times2=8$. (c) (1,T),(3,T) → $\frac{2}{8}=\frac{1}{4}$. (d) (3,H),(3,T),(4,H),(4,T) → $\frac{4}{8}=\frac{1}{2}$.
- (b) (2,5)=10 → $\frac{1}{6}$. (c) Products 4,8,10,12 → $\frac{4}{6}=\frac{2}{3}$.
- (b) $30-10=\mathbf{20}$. (c) $50-45=\mathbf{5}$. (d) $\frac{15}{50}=\frac{3}{10}$.
- (b) $3\times2\times2=\mathbf{12}$. (c) (Soup,Chicken,Cake) and (Soup,Beef,Cake) → $\frac{2}{12}=\frac{1}{6}$.
FAQs
Q: Do I have to draw the whole table in the exam?
A: Yes — if the question says “construct a possibility space”, the complete table with all outcomes must be shown. That is where your method marks come from.
Q: Can I use a tree diagram instead of a table?
A: Tree diagrams are valid for sequential events, but Foundation questions often ask for a two-way table specifically. If the method is your choice, use a table for two combined events.
Q: Must I always simplify my probability fraction?
A: Yes, always simplify to lowest terms in your final answer unless the question says otherwise.
Need Help with Probability?
A qualified GCSE maths tutor can guide you through possibility spaces, Venn diagrams, tree diagrams, and every other probability topic on the syllabus.
Find a GCSE Maths Tutor
