GCSE Maths Practice: probability-scale

Understanding Probability for Numbers Less Than a Value

This question focuses on finding the probability of rolling a number less than 3 on a fair 6-sided die. These types of questions are common in GCSE Foundation Maths because they help build the essential skill of identifying favourable outcomes based on a condition. Instead of looking for one specific number, the question asks for all numbers that meet a requirement. This improves reasoning, counting skills, and confidence when dealing with probability involving multiple outcomes.

The Structure of a Fair Die

A standard die has the following six outcomes: 1, 2, 3, 4, 5, and 6. Each number is equally likely to appear when the die is rolled. Because of this fairness, each individual outcome has a probability of one out of six.

When a condition such as “less than 3” is used, we need to list all numbers that satisfy the condition before forming the probability fraction.

Step-by-Step Method

1. Write the sample space: {1, 2, 3, 4, 5, 6}.
2. Identify which numbers are less than 3. These are 1 and 2.
3. Count favourable outcomes: 2 numbers satisfy the condition.
4. Count total outcomes: a fair die always has 6 outcomes.
5. Write the probability as a fraction: favourable ÷ total.

Worked Example 1: Rolling a Number Less Than 5

Numbers less than 5 are 1, 2, 3, and 4. There are 4 favourable outcomes. Using the same process, the probability becomes 4 out of 6.

Worked Example 2: Rolling a Number More Than 2

Numbers greater than 2 are 3, 4, 5, and 6. That means there are also 4 favourable outcomes. The probability is again 4 out of 6. Even though the numbers change, the method stays consistent.

Worked Example 3: Rolling an Odd Number

The odd numbers on a standard die are 1, 3, and 5. That gives 3 favourable outcomes out of 6 total outcomes. Forming the probability gives 3/6, which you may simplify.

Common Mistakes to Avoid

• Including the boundary number: “Less than 3” does not include 3. Only the numbers 1 and 2 count.
• Miscounting favourable outcomes: Always list the numbers first to avoid skipping any.
• Thinking previous rolls matter: Each roll is independent. Earlier outcomes do not change the next probability.
• Using decimals or whole numbers unnecessarily: Probability should usually be expressed as a fraction at GCSE level unless the question asks otherwise.

Why Conditional Probability Matters

Many GCSE questions ask for probabilities involving conditions such as “greater than”, “less than”, “odd”, “even”, or “between two values”. Mastering how to identify favourable outcomes quickly makes those questions much easier. This skill also prepares students for handling probability trees, multi-step events, and more complex combined outcomes later in the course.

Real-Life Connections

Conditional probability appears in real life in situations like predicting weather, analysing data, estimating risks, and making decisions based on limited information. Whenever we narrow down a group to only the options that meet certain criteria, we are applying the same thinking used in this question.

Frequently Asked Questions

Q1: What does “less than” mean in maths?
It means all numbers smaller than the given number. It does not include the number itself.

Q2: Can the probability change after several rolls?
No. A fair die does not change behaviour. Each roll is independent.

Q3: Should the answer be simplified?
Simplifying is optional unless the question asks for it. Both forms are mathematically valid.

Study Tip

Whenever you see conditions like “less than” or “greater than”, write down the numbers that satisfy the condition before doing anything else. This ensures accuracy and reduces mistakes.