GCSE Maths Practice: listing-outcomes

Understanding Probability on a Fair Six-Sided Die

This question focuses on a very common GCSE Foundation probability skill: identifying favourable outcomes on a six-sided die and expressing the probability as a simplified fraction. A standard die has six faces numbered from 1 to 6. When you are asked to find the probability of rolling a number greater than a certain value, you must list all the numbers that satisfy the condition and then compare them with the total number of possible outcomes.

In probability, every outcome on a fair die is equally likely because each face has the same chance of landing face-up. This idea of equal likelihood is essential: it ensures that we simply count outcomes rather than needing more advanced techniques.

Step-by-Step Method

1. Write down all numbers on a standard die: 1, 2, 3, 4, 5, and 6.
2. Identify which of these numbers are greater than 4. Only the numbers that meet the condition count as favourable outcomes.
3. Count how many favourable numbers there are.
4. Write the probability as a fraction: favourable outcomes ÷ total outcomes.
5. Simplify the fraction if possible.

This structured approach works for any similar problem involving inequalities and discrete outcomes.

Worked Example 1: Rolling a Number Less Than 3

The numbers less than 3 on a die are 1 and 2. That gives 2 favourable outcomes out of 6. The probability is 2/6, which simplifies to 1/3. Even though the condition has changed, the process remains identical.

Worked Example 2: Rolling an Even Number

The even numbers on a die are 2, 4, and 6. That gives 3 favourable outcomes. The probability is 3/6, which simplifies to 1/2. This example highlights the importance of knowing basic number properties when answering probability questions.

Worked Example 3: Rolling a Number Between 2 and 5

The numbers that satisfy this condition are 3, 4, and 5. So there are 3 favourable outcomes out of 6, giving a probability of 3/6 = 1/2. This shows how inequalities can define different sets of favourable outcomes.

Common Mistakes

• Including numbers that do not meet the condition. For example, some students accidentally include 4 when the condition is “greater than 4”.
• Incorrect total count. A fair die always has 6 outcomes.
• Not simplifying fractions. GCSE exam questions often expect the simplified form, especially at Foundation level.
• Confusing probability with counting. Probability requires dividing favourable outcomes by the total, not just listing possibilities.

Real-Life Applications

Although dice problems seem abstract, the underlying probability ideas appear in real-life decision-making, games, and even technology. Board games rely heavily on dice probability to create fairness and unpredictability. In computing, random number generators simulate dice-like behaviour when creating digital randomness for games, testing, and encryption. In science experiments, understanding outcomes helps in predicting results and interpreting data.

FAQ

Q: Why do we only count numbers that meet the condition?
A: Probability focuses on the outcomes relevant to the event of interest, known as favourable outcomes.

Q: Does the order matter on a single die roll?
A: No. A single roll produces one number, so order does not play a role.

Q: Can probability be greater than 1?
A: Never. Probabilities always fall between 0 and 1.

Study Tip

Before forming a fraction, always list the favourable outcomes in full. This prevents overlooking values and ensures your probability calculation is accurate.