GCSE Maths Practice: listing-outcomes

Question 5 of 10

Practise calculating probabilities involving number properties on a die.

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\( \begin{array}{l}\textbf{What is the probability of rolling an} \\ \textbf{even number on a fair 6-sided die?}\end{array} \)

Choose one option:

Identify number properties clearly before forming the fraction.

Understanding Probability When Rolling an Even Number

This question explores one of the simplest and most important ideas in GCSE Foundation probability: identifying favourable outcomes on a fair six-sided die. A standard die has six faces labelled 1 to 6. Because the die is fair, each number has an equal chance of appearing, which makes this type of question ideal for building confidence with probability fractions.

The event described here is rolling an even number. Even numbers are whole numbers that can be divided exactly by 2. On a six-sided die, the even numbers are 2, 4, and 6. Identifying these correctly is the key first step. Once you know which outcomes are favourable, the probability becomes a simple matter of forming a fraction using the rule:

Probability = favourable outcomes ÷ total outcomes

Step-by-Step Method

  1. Write down all possible numbers on the die: 1, 2, 3, 4, 5, 6.
  2. Identify the favourable outcomes — the even numbers: 2, 4, and 6.
  3. Count how many favourable outcomes there are: 3.
  4. Count the total number of outcomes: 6.
  5. Form the probability as a fraction: 3 ÷ 6 = 3/6.
  6. Simplify the fraction by dividing numerator and denominator by 3: 1/2.

These steps form a reliable method for solving any similar problem that asks for the probability of choosing or rolling numbers with specific properties, such as “odd numbers”, “numbers greater than 3”, or “multiples of 2”.

Worked Example 1: Rolling an Odd Number

The odd numbers on a die are 1, 3, and 5 — three favourable outcomes. Therefore, the probability of rolling an odd number is also 3/6 = 1/2.

Worked Example 2: Rolling a Multiple of 3

The multiples of 3 from 1 to 6 are 3 and 6. This gives 2 favourable outcomes. The probability is 2/6, which simplifies to 1/3.

Worked Example 3: Rolling a Number Greater Than 4

The numbers greater than 4 are 5 and 6. That gives 2 favourable outcomes out of 6, so the probability is 2/6 = 1/3.

Common Mistakes to Avoid

  • Incorrectly identifying even numbers. Sometimes students include 3 or forget 6 — always check which numbers are divisible by 2.
  • Not simplifying the final fraction. While 3/6 is correct, examiners prefer simplified fractions like 1/2.
  • Mixing up number properties. For example, multiples of 2 (even numbers) are not the same as multiples of 3.
  • Assuming the die is biased. Unless stated otherwise, all outcomes are equally likely.

Real-Life Applications

Understanding simple dice probability helps in many everyday and analytical situations. Board games often rely on dice for movement or outcomes. Probability concepts also appear in computer game design, simulations, risk assessment, and scientific investigations. Even machine learning and data modelling rely on probability distributions at their core, although at a much higher level. Mastering the basics now builds a strong foundation for future study.

FAQ

Q: Why do we divide by 6?
A: A fair die has six equally likely outcomes, so probability is based on all six possibilities.

Q: Do we count each even number separately?
A: Yes — each number represents a unique outcome.

Q: Can probability ever be greater than 1?
A: No. Probabilities always range from 0 to 1.

Study Tip

When a probability question mentions a number property (such as even, odd, prime, multiple, or factor), list all numbers on the die first. Then filter only the ones that match the property — this prevents mistakes and guarantees an accurate probability.

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