GCSE Maths Practice: listing-outcomes

Understanding Combined Outcomes in GCSE Foundation Probability

In Foundation level GCSE Maths, one of the key skills in probability is learning how to count all the possible outcomes when two events happen together. In this case, the two events are rolling a die and flipping a coin. Even though the operations seem simple, this idea is extremely important because it forms the basis of later probability topics such as outcome grids, sample spaces, probability tables, and tree diagrams.

When two events happen at the same time, we want to consider every possible combination. A die roll can land on any of its six faces, while a coin can land on heads or tails. Since the result of one does not affect the other, the events are classed as independent. This means we multiply the number of outcomes to find the combined total. Mastering this method will help you confidently tackle more complex questions later in the GCSE course.

How Independent Events Work

Two events are independent if the result of the first event does not influence or limit the result of the second. Rolling a die does not change how the coin behaves. Regardless of which number appears on the die, the coin still has two outcomes. In everyday language, the events do not interfere with each other.

Because the events are independent, we use multiplication to combine their outcome counts. This multiplication principle appears again when you study probability trees, replacement and non-replacement problems, and compound probability questions.

Step-by-Step Method

1. Identify the number of outcomes for the first event (rolling a 6-sided die gives 6 outcomes).
2. Identify the number of outcomes for the second event (flipping a coin gives 2 outcomes).
3. Multiply the numbers together: number of die outcomes × number of coin outcomes.
4. The result gives the size of the sample space (total combinations).

Worked Example 1: Rolling a Die and Choosing a Colour

Imagine you roll a die (6 outcomes) and then randomly choose a colour card from a set of 3 colours. To find the total possible pairs, multiply:
6 × 3 = 18 outcomes. Examples could be (1, Red), (4, Blue), (6, Green), and so on.

Worked Example 2: Spinning a Spinner and Flipping a Coin

If a spinner has 4 sections and you flip a coin, you multiply:
4 × 2 = 8 possible outcomes. Listing them systematically ensures none are missed: (1,H), (1,T), (2,H), (2,T), etc.

Worked Example 3: Two Different Dice

Suppose you roll a red die and a blue die. Each has 6 outcomes, so the total number of combined outcomes is:
6 × 6 = 36. This is the principle behind multiplication grids used in probability questions involving sums of dice.

Common Mistakes to Avoid

• Adding instead of multiplying. Students often try 6 + 2, which counts events separately instead of combining them.
• Forgetting independence. Some learners think the coin outcome depends on the die result, but it does not.
• Missing combinations when listing outcomes manually. A structured list or table prevents this.

Real-Life Applications

This skill appears in science experiments, board game design, computer simulations, and any situation where two random events combine. For example, video games often use combined random systems to determine outcomes such as item drops or enemy behaviour. In statistics and data science, independent event counting helps when building probability models.

FAQ

Q: Why do we multiply instead of add?
A: Multiplication counts every combination of the two events, not just the events individually.

Q: Does the order matter?
A: Yes—(die roll 3, heads) is a different outcome from (die roll 3, tails).

Q: If the die had 10 sides, would the method change?
A: No, you still multiply. It would be 10 × 2.

Study Tip

Whenever dealing with combined events, draw a quick table or grid. This helps you visualise all outcomes clearly and reduces mistakes.