GCSE Maths Practice: probability-scale

Understanding Probability When Looking for Patterns

This question focuses on finding the probability of rolling an even number on a fair 6-sided die. This type of probability task strengthens pattern recognition skills, reinforces understanding of simple sample spaces, and prepares students for more complex probability concepts. In GCSE Foundation Maths, problems like this appear often because they require identifying more than one favourable outcome, not just a single value.

The Structure of a Fair 6-Sided Die

A standard die includes the numbers 1, 2, 3, 4, 5, and 6. All outcomes are equally likely because the die is fair. When asked for the probability of rolling an even number, we look for numbers in the sample space that match the condition of being even. The even numbers are 2, 4, and 6, which gives us three favourable outcomes.

Step-by-Step Strategy

1. Write the sample space: {1, 2, 3, 4, 5, 6}.
2. Identify numbers that match the condition “even”: these are 2, 4, and 6.
3. Count favourable outcomes: there are 3.
4. Count total outcomes: 6 on a fair die.
5. Form the probability fraction: favourable ÷ total.

Worked Example 1: Probability of Rolling an Odd Number

The odd numbers on a die are 1, 3, and 5. There are three favourable outcomes out of six possible outcomes, so the probability becomes 3/6. This mirrors the method used for even numbers.

Worked Example 2: Probability of Rolling a Multiple of 3

The multiples of 3 on a die are 3 and 6. That gives two favourable outcomes. With six possible outcomes, the probability becomes 2/6.

Worked Example 3: Probability of Rolling a Prime Number

Prime numbers on a die are 2, 3, and 5. That gives three favourable outcomes. Forming the fraction results in 3/6. These examples show how the same reasoning applies to different number patterns.

Common Misunderstandings

• Not listing the sample space: Students may skip listing numbers, causing them to miss one of the favourable outcomes.
• Including incorrect numbers: Some believe 1 is even or that 0 appears on a die. Writing the sample space avoids these mistakes.
• Assuming previous rolls matter: Probability stays the same for each roll because the events are independent.
• Thinking probability must be simplified: Simplification is optional unless specified.

Real-Life Context

Understanding patterns like even and odd numbers is important beyond exam questions. Many games, statistics tasks, and logical decisions involve identifying categories of outcomes. For example, when analysing experimental data or designing fair games, recognising symmetrical patterns ensures decisions are based on accurate reasoning.

Why This Skill Is Important

This type of conditional probability builds the foundation for working with combined events, probability trees, and more advanced mathematical reasoning. It improves confidence in recognising favourable outcomes and strengthens fraction fluency. The ability to work efficiently with small sets of outcomes also supports later GCSE topics such as sampling, expected frequencies, and events with equal likelihood.

Frequently Asked Questions

Q1: Why are there 3 even numbers?
Even numbers are those divisible by 2. On a die, these are 2, 4, and 6.

Q2: Do we have to simplify 3/6?
Simplification is allowed (to 1/2) but not required unless instructed.

Q3: Does rolling many times change the probability?
No. Each roll is independent, meaning previous rolls do not affect future outcomes.

Study Tip

Whenever a question asks for numbers with a property (even, odd, prime, greater than, etc.), always write the sample space and circle the matching values. This avoids errors and speeds up working time.