GCSE Maths Practice: probability-basics

Understanding Probability with Numbers on a Die

Dice-based probability problems are common in GCSE Maths because they use simple, equally likely outcomes. A fair six-sided die has faces numbered from 1 to 6, and each face has the same chance of appearing when the die is rolled. This equal likelihood makes it straightforward to apply the basic probability formula and identify favourable outcomes.

Identifying Favourable Outcomes

When a question asks for the probability of rolling a number greater than 4, it means you need to look at all numbers on the die and pick those that satisfy the condition. The numbers greater than 4 are 5 and 6. That gives two favourable outcomes. The remaining numbers—1, 2, 3 and 4—do not meet the condition and therefore do not count towards the favourable outcomes.

Total Possible Outcomes

Since the die has six faces, there are six total possible outcomes. In probability language, this total is called the sample space. Every calculation begins by identifying the size of the sample space because probability involves comparing the favourable outcomes to the total number of possible outcomes.

The Probability Formula

Use the GCSE standard probability formula:

Probability = (Favourable outcomes) ÷ (Total outcomes)

In this scenario, there are 2 favourable outcomes and 6 total outcomes, giving 2/6.

Worked Example 1: Rolling an Even Number

The even numbers on a die are 2, 4 and 6. That gives 3 favourable outcomes. The total number of outcomes remains 6, so the probability is 3/6, which simplifies to 1/2. This example demonstrates how the number of favourable outcomes changes depending on the condition.

Worked Example 2: Rolling a Number Less Than 3

The numbers less than 3 are 1 and 2. That yields 2 favourable outcomes out of 6, giving a probability of 2/6, again simplifying to 1/3. This mirrors the same reasoning used in the original question but uses a different condition.

Worked Example 3: Rolling a Prime Number

The prime numbers on a die are 2, 3 and 5. That gives 3 favourable outcomes. Using the formula gives 3/6, which simplifies to 1/2. This is a useful example because it helps reinforce identification skills needed in probability questions involving number properties.

Common Mistakes

• Forgetting that the die has exactly six outcomes—no more, no fewer.
• Including numbers that do not satisfy the condition, such as counting 4 by mistake.
• Misinterpreting the phrase “greater than” and confusing it with “greater than or equal to.”

Real-Life Applications

Understanding simple probability builds the foundation for more advanced ideas used in real-world situations such as risk analysis, statistics, simulations and game design. Dice models are also used in computer science when creating random number generators or testing algorithms. The ability to calculate probabilities quickly and accurately is a valuable skill across many fields.

FAQ

Q: Does rolling the die more times change the probability?
No. Each roll is independent, meaning previous results do not change future probabilities.

Q: Should the fraction be simplified?
You may simplify 2/6 to 1/3 if a simplified form is expected, but both forms are mathematically correct unless the question states otherwise.

Q: Are all dice guaranteed to be fair?
In GCSE Maths, yes. All dice are assumed fair unless the question specifies otherwise.

Study Tip

When working with dice, always list the full set of outcomes first: 1, 2, 3, 4, 5, 6. Then pick only those that match the condition. This approach helps ensure accuracy and prevents missing any values.