GCSE Maths Practice: probability-basics

Question 7 of 11

Work with two-dice probability and sums of outcomes.

‹ ›

\( \begin{array}{l}\textbf{Two dice are rolled.} \\ \textbf{What is the probability} \\ \textbf{that their sum is 7?}\end{array} \)

Choose one option:

List every pair of dice outcomes that make the required sum.

Understanding Probability with Two Dice

Two-dice probability is a key part of higher-tier GCSE Maths because it introduces ideas such as combined outcomes, sample-space grids and symmetrical distributions. When rolling two fair dice, each die has six faces, so there are 6 × 6 = 36 equally likely ordered pairs. This clear structure allows you to calculate probabilities for specific sums, compare likelihoods and identify patterns across the sample space.

The Sample Space for Two Dice

The sample space consists of all ordered pairs (a, b) where a and b are numbers from 1 to 6. For example: (1,1), (1,2), (2,1), and so on. Because each die is independent and fair, each of the 36 outcomes has the same probability. Understanding this grid is essential for solving sum-based probability problems such as finding the likelihood of rolling a 7.

Why the Sum of 7 Is Special

The sum of 7 is often used in GCSE Maths because it has the highest number of combinations among all possible sums. These combinations are:

(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)

The symmetry makes the total number of favourable outcomes easy to recognise. There are six such pairs, which is more than any other sum.

Worked Example 1: Sum of 10

Pairs that sum to 10 are (4,6), (5,5) and (6,4). That gives three favourable outcomes. The probability is therefore 3/36, which simplifies to 1/12. This process mirrors the method used for sum of 7 but with fewer combinations.

Worked Example 2: Sum of 2 or Sum of 12

These are the least likely sums because each has only one combination:

  • Sum of 2 → (1,1)
  • Sum of 12 → (6,6)

Each has probability 1/36.

Worked Example 3: Sum Greater Than 8

To find the probability of rolling a sum greater than 8, list all pairs: 9 (4 combinations), 10 (3 combinations), 11 (2 combinations), 12 (1 combination). Total favourable outcomes = 10. Probability = 10/36, simplified to 5/18. This example builds on the same method and develops higher-tier reasoning.

Common Mistakes

  • Assuming the sums are equally likely—even though combinations differ.
  • Forgetting that the pairs are ordered, meaning (2,5) and (5,2) are different outcomes.
  • Incorrectly multiplying probabilities instead of listing outcomes when sums are involved.
  • Thinking two dice have 12 outcomes instead of 36—this leads to major errors.

Real-Life Applications

Two-dice probability is widely used in gaming, simulation modelling, statistical reasoning and random event generation in computing. Understanding the structure of combined outcomes helps with more advanced topics such as probability distributions, expected values and statistical sampling. It also forms a foundation for A-level Mathematics and Statistics modules.

FAQ

Q: Why are some sums more likely than others?
Because different sums have different numbers of outcome pairs. The more combinations, the more likely the sum.

Q: Are the dice independent?
Yes. The outcome of one die does not affect the other, which is why every pair has equal probability.

Q: Should we simplify 6/36?
You may simplify to 1/6 if the question asks for simplest form, but 6/36 is acceptable unless stated.

Study Tip

Whenever handling two-dice probability, sketch a quick 6 × 6 table. This makes it easy to visualise all 36 outcomes, identify symmetrical patterns and avoid missing any combinations. This technique is excellent for GCSE exam accuracy and speed.

Related Quizzes