GCSE Maths Practice: listing-outcomes

Understanding Probability on a Six-Sided Die

This question focuses on a key GCSE Foundation probability skill: identifying the number of outcomes that satisfy a given condition and expressing the result as a simplified fraction. The condition here is “rolling a number less than or equal to 4”. Because a standard die has six equally likely outcomes, this type of problem is straightforward once you know how to identify favourable outcomes.

A six-sided die (often called a d6) has the numbers 1 to 6. Each face has an equal chance of landing face-up, meaning probability depends only on counting outcomes rather than weighting them. The phrase “less than or equal to 4” includes all numbers from 1 to 4, so the range is clear and easy to list.

Step-by-Step Method

1. Write out all possible numbers: 1, 2, 3, 4, 5, 6.
2. Identify which numbers satisfy the condition “less than or equal to 4”. These are 1, 2, 3, and 4.
3. Count the favourable outcomes: there are 4 numbers.
4. Count all possible outcomes: the die has 6 numbers.
5. Form the probability fraction: favourable ÷ total = 4/6.
6. Simplify the fraction by dividing top and bottom by 2: 4/6 becomes 2/3.

This step-by-step structure can be applied to almost any simple die-based probability question encountered at GCSE Foundation level.

Worked Example 1: Rolling a Number Greater Than 2

The numbers greater than 2 are 3, 4, 5, and 6. That gives 4 favourable outcomes out of 6, so the probability is 4/6 = 2/3. This shows how the method stays the same even when conditions change.

Worked Example 2: Rolling a Number Less Than 3

The numbers less than 3 are 1 and 2. That gives 2 favourable outcomes. The probability is 2/6, which simplifies to 1/3. This also demonstrates the importance of simplifying your answer.

Worked Example 3: Rolling a Number Between 2 and 5

The numbers between 2 and 5 (inclusive) are 2, 3, 4, and 5. That gives 4 favourable outcomes, leading to a probability of 4/6 = 2/3. Again, the counting approach remains identical.

Common Mistakes

• Incorrectly identifying the range. Some students mistakenly include 5 because they misread “less than or equal to 4”.
• Forgetting to include 4 in the favourable set. The phrase “or equal to” is easy to overlook.
• Not simplifying fractions. Examiners often expect answers like 2/3 rather than 4/6.
• Thinking a die has numbers 0–5. A standard die always has 1–6 unless stated otherwise.

Real-Life Applications

This type of probability appears frequently in games, simulations, and decision-making. Board games rely on dice probabilities to determine movement and outcomes. In computing, random number generation uses similar principles to simulate dice rolls. In scientific experiments or surveys, the idea of counting favourable outcomes appears when analysing categorical data or predicting results.

FAQ

Q: Why do we divide by 6?
A: There are 6 possible outcomes on a fair die, and all are equally likely.

Q: Do we have to list the numbers every time?
A: At Foundation level, listing them helps to avoid mistakes, especially with inequalities.

Q: Why do we simplify the fraction?
A: Simplified fractions are standard in GCSE assessments and show clear understanding.

Study Tip

Whenever a question uses terms like “less than”, “greater than”, or “between”, rewrite the condition with the exact numbers included. This prevents misunderstanding and makes the probability calculation much easier.