Listing Outcomes Quizzes

Introduction

Listing outcomes is a fundamental skill in GCSE Maths probability. It involves identifying all possible results of an experiment, which is essential for calculating probabilities accurately. By systematically listing outcomes, students can visualise the sample space, avoid missing possibilities, and solve more complex probability problems with confidence.

Core Concepts

What Does "Listing Outcomes" Mean?

Listing outcomes means writing down every possible result of an experiment. This creates the sample space, which is the foundation for calculating probabilities of events.

• Experiment: Any action that produces results (e.g., rolling a die, tossing a coin).
• Outcome: A single possible result of an experiment.
• Event: One or more outcomes of interest (e.g., rolling an even number).
• Sample Space: The set of all possible outcomes.

Why Listing Outcomes is Important

• Ensures all possibilities are considered before calculating probabilities.
• Helps avoid double-counting or missing outcomes.
• Essential for combined events, such as tossing two coins or rolling two dice.
• Supports understanding of tree diagrams, Venn diagrams, and other probability tools.

Rules & Steps for Listing Outcomes

1. Identify the experiment and the variables involved.
2. Determine the possible outcomes for each variable.
3. Use systematic methods to combine outcomes:
   • For single-step experiments, list all outcomes directly.
   • For multi-step experiments, use tables, lists, or Cartesian products.
4. Ensure each outcome is listed exactly once to avoid duplication.
5. Verify the total number of outcomes matches expectations (multiply numbers of options for each step in combined events).

Worked Examples

Example 1: Single Event

Experiment: Toss a single coin.

Step 1: Identify outcomes: Heads (H) or Tails (T)

Sample space: \( S = \{H, T\} \)

Step 2: Probability of getting heads:

$$ P(\text{H}) = \frac{1}{2} = 0.5 $$

Example 2: Two Coin Tosses

Experiment: Toss two coins.

Step 1: Identify outcomes for each coin: H or T

Step 2: List all combined outcomes systematically:

• HH, HT, TH, TT

Step 3: Event: Exactly one head → outcomes = {HT, TH}

Step 4: Probability:

$$ P(\text{exactly one head}) = \frac{2}{4} = \frac{1}{2} = 50\% $$

Example 3: Rolling Two Dice

Experiment: Roll two six-sided dice.

Step 1: List outcomes systematically using ordered pairs (Die1, Die2):

• (1,1), (1,2), (1,3), (1,4), (1,5), (1,6)
• (2,1), (2,2), (2,3), …, (2,6)
• …
• (6,1), (6,2), …, (6,6)

Total outcomes = 6 × 6 = 36

Step 2: Event: Sum of dice = 7 → outcomes = (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)

$$ P(\text{sum = 7}) = \frac{6}{36} = \frac{1}{6} \approx 16.67\% $$

Example 4: Using a Table to List Outcomes

Experiment: Roll a die and flip a coin.

Step 1: List die outcomes: 1, 2, 3, 4, 5, 6

Step 2: List coin outcomes: H, T

Step 3: Create a table:

	H	T
1	(1,H)	(1,T)
2	(2,H)	(2,T)
3	(3,H)	(3,T)
4	(4,H)	(4,T)
5	(5,H)	(5,T)
6	(6,H)	(6,T)

Total outcomes = 6 × 2 = 12

Event: Die shows even and coin shows heads → outcomes = (2,H), (4,H), (6,H)

$$ P(\text{even & heads}) = \frac{3}{12} = \frac{1}{4} = 25\% $$

Example 5: Three-Step Experiment

Experiment: Draw a card from a 4-card deck (A, B, C, D), roll a coin, and toss a die (1–3).

Step 1: List card outcomes: A, B, C, D

Step 2: Coin outcomes: H, T

Step 3: Die outcomes: 1, 2, 3

Step 4: Combine systematically: 4 × 2 × 3 = 24 outcomes

Step 5: Event: Card = A, coin = H → outcomes = (A,H,1), (A,H,2), (A,H,3)

$$ P(\text{Card A & H}) = \frac{3}{24} = \frac{1}{8} = 12.5\% $$

Common Mistakes

• Missing some outcomes, especially in multi-step experiments.
• Double-counting outcomes.
• Not multiplying the number of options for each step correctly in combined experiments.
• Confusing ordered vs unordered outcomes (e.g., coin toss sequence matters: HT ≠ TH).
• Using incorrect notation or inconsistent labelling.

Applications

Listing outcomes is used in exams and real-world probability analysis:

• Games: Calculating chances in dice, cards, and coin games.
• Probability trees: Base for creating tree diagrams for multi-step events.
• Sampling: Understanding possible combinations of items or people.
• Risk assessment: Identifying all possible scenarios in decision-making.

Strategies & Tips

• Always start systematically: list one variable at a time and combine outcomes step by step.
• Use tables, diagrams, or ordered pairs to organise outcomes clearly.
• Check your total outcomes against the expected product of options for each step.
• When events are independent, multiply probabilities of individual outcomes to find combined probability.
• Label events clearly to avoid confusion, especially in multi-step experiments.
• Practice with coins, dice, cards, and multi-step scenarios to build confidence.

Summary & Encouragement

Listing outcomes is the foundation of probability calculations. Key points to remember:

• Identify the experiment and its sample space.
• Systematically list all possible outcomes to avoid missing or double-counting.
• Use tables, ordered pairs, or diagrams for clarity, especially for multi-step experiments.
• Calculate probabilities accurately using favourable outcomes divided by total outcomes.
• Practice a variety of single-step and combined events to strengthen your understanding.

Work through examples with coins, dice, cards, and multi-step experiments. This will improve your ability to calculate probabilities accurately and enhance your performance in GCSE Maths. Complete the quizzes to reinforce your knowledge of listing outcomes!