Listing Outcomes Systematically
Ever tried to work out how many different outfits you can make, or how many meal combinations a restaurant offers? In maths, being able to list all possible outcomes in a clear, systematic way is a key skill for understanding probability.
Being systematic means following a plan to find every single possibility without errors. Let’s explore the methods you can use.
Key Terms and Vocabulary
Outcome
A single, possible result of an event. For example, “Heads” is one outcome of flipping a coin.
Systematic
Following a fixed plan or logical method to ensure you don’t miss any outcomes or count any twice.
Combination
The different ways items can be put together. Listing all outcomes means listing all possible combinations.
Sample Space
The complete set of all possible outcomes of an event. A systematic list creates the sample space.
Worked Examples
Example 1: Tree Diagram 🌳
A tree diagram is great for visualising branching choices. Imagine you have two tops and two pairs of bottoms:
- Tops: Red, Blue
- Bottoms: Jeans, Trousers
By following each path, you find the 4 possible outfits.
Example 2: Two-Way Table 🍽️
Two-way tables are perfect for pairing two sets of choices. A café offers:
- Mains: Pizza, Pasta
- Drinks: Cola, Lemonade, Water
| 🥤 Cola | 🍋 Lemonade | 💧 Water | |
|---|---|---|---|
| 🍕 Pizza | Pizza, Cola | Pizza, Lemonade | Pizza, Water |
| 🍝 Pasta | Pasta, Cola | Pasta, Lemonade | Pasta, Water |
Counting the cells gives you 6 combinations. You can also do a quick check: 2 Mains × 3 Drinks = 6.
Example 3: Systematic List 🍦
A well-organised list is often clearest. An ice cream shop has:
- Flavours: Vanilla, Chocolate, Strawberry
- Toppings: Sprinkles, Sauce
Start with Vanilla:
• Vanilla & Sprinkles
• Vanilla & Sauce
Now do Chocolate:
• Chocolate & Sprinkles
• Chocolate & Sauce
Finally, Strawberry:
• Strawberry & Sprinkles
• Strawberry & Sauce
Counting them up gives you 6 combinations.
Tutor Insights
🤔 Common Misunderstandings
- Not being systematic: Brainstorming answers randomly often leads to missed outcomes or repetitions. We always remind students to “stick to the system!”
- Miscounting: Even with a perfect list, it’s easy to miscount the total. Always double-check your final number.
📝 Common Exam Mistakes
- Incomplete lists or tables: This is the biggest reason for losing marks on these questions.
- Poor presentation: A messy list is hard for you to check and hard for an examiner to mark.
- Not showing working: If the question asks you to list the outcomes, just writing the total number isn’t enough.
Practice Questions
Here are some questions to help you master listing combinations systematically.
- A coin is flipped, and a six-sided dice is rolled. List all possible outcomes.
- A café offers three sandwiches: Cheese (C), Ham (H), and Chicken (K), and two types of crisps: Salted (S) and Vinegar (V).
a) Create a two-way table for all combinations.
b) How many combinations are there? - You can choose one fruit: Apple (A), Banana (B), or Orange (O), and one liquid: Milk (M) or Water (W). List all smoothie combinations.
- A car is available in four colours: Red (R), Blue (B), Green (G), White (W), and with two doors (2D) or four doors (4D). List all different models.
Show Answers
- 12 outcomes: (H,1), (H,2), (H,3), (H,4), (H,5), (H,6) and (T,1), (T,2), (T,3), (T,4), (T,5), (T,6).
-
a) Table:
b) 6 combinations.
Salted Vinegar Cheese (C, S) (C, V) Ham (H, S) (H, V) Chicken (K, S) (K, V) - 6 combinations: (A,M), (A,W), (B,M), (B,W), (O,M), (O,W).
- 8 combinations: (R,2D), (R,4D), (B,2D), (B,4D), (G,2D), (G,4D), (W,2D), (W,4D).
FAQs
Q: When should I use a table versus a list?
A: Two-way tables are brilliant when you have exactly two categories of choices (e.g., mains and drinks). If you have three or more categories (e.g., starter, main, dessert), a structured list is often easier to manage.
Q: What if there are too many outcomes to list?
A: For GCSE Foundation, questions are usually designed so you can list them all. To check your answer, you can use the multiplication principle: $\text{choices for A} \times \text{choices for B}$. For example, 3 tops and 4 bottoms give $3 \times 4 = 12$ combinations.
Want to Tackle Probability with Confidence?
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