GCSE Maths Practice: probability-scale

Understanding Probability with Conditions

Some probability questions involve looking for numbers that match a particular condition rather than a single value. In this case, we are searching for numbers greater than 4 on a fair 6-sided die. Conditional events like this appear frequently in GCSE Foundation Maths and build important skills for later probability topics.

The Structure of a Fair Die

A standard die has six equally likely outcomes: 1, 2, 3, 4, 5, and 6. Because the die is fair, each number has an equal probability of being rolled. This means that one out of six is always the starting point for single-number probability. When conditions involve more than one possible number, the method is the same — you simply count how many values satisfy the condition.

Step-by-Step Method

1. List the sample space: {1, 2, 3, 4, 5, 6}.
2. Identify which numbers meet the condition “greater than 4”. These are 5 and 6.
3. Count the favourable outcomes: 2.
4. Count the total outcomes: 6.
5. Form the probability: favourable ÷ total.

Worked Example 1: Rolling a Number Less Than 3

The values less than 3 are 1 and 2. That means two favourable outcomes. With six outcomes in total, the probability is 2/6 using the same method as this question.

Worked Example 2: Rolling an Even Number

The even numbers are 2, 4, and 6. That gives three favourable outcomes. The total outcomes remain six, so the probability becomes 3/6. This can be simplified, but the key skill is identifying the favourable outcomes correctly.

Worked Example 3: Rolling a Number Greater Than 2

Numbers greater than 2 are 3, 4, 5, and 6. That means four favourable outcomes. Writing the probability gives 4/6. Again, the method remains consistent.

Common Mistakes to Avoid

• Confusing the condition: Some learners accidentally include the boundary number (4) even though the question says “greater than 4”. Always read conditions carefully.
• Not listing the sample space: Skipping this step can cause you to miscount favourable outcomes.
• Thinking probability changes after several rolls: Each roll of a fair die is independent. Earlier rolls do not affect later ones.
• Forgetting that only whole numbers appear on dice: Sometimes students include values like 4.5 by mistake.

Real-Life Applications

Understanding conditional probability helps in everyday decision-making. For example, when predicting the likelihood of events in games, assessing risk, or analysing scientific results, similar reasoning is required. The idea of counting how many outcomes meet a condition is central to many real-world problems involving uncertainty.

Why This Skill Matters

GCSE exams often include questions where more than one outcome is acceptable. Learning to quickly identify favourable outcomes makes more advanced topics easier, such as probability trees, combined events, sampling, and independent outcomes. It also builds confidence in handling fractions and logical reasoning.

Frequently Asked Questions

Q1: Do conditions like “greater than” always exclude the boundary?
Yes. “Greater than 4” means only numbers above 4. If the question wanted to include 4, it would say “greater than or equal to”.

Q2: Does the order of outcomes matter?
No. A single roll gives only one result. Probability depends only on counting outcomes, not on order.

Q3: Is simplifying the fraction required?
You may simplify, but the original fraction is acceptable unless the question asks for simplification.

Study Tip

Whenever a condition appears in a probability question, quickly write the sample space and circle or underline the values that meet the condition. This reduces errors and speeds up problem-solving.