GCSE Maths Practice: probability-scale

Question 9 of 10

Find the probability that two dice rolled together produce a sum of 8.

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\( \begin{array}{l}\textbf{What is the probability of rolling} \\ \text{a sum of 8 using two fair} \\ \text{6-sided dice?}\end{array} \)

Choose one option:

List every pair that totals 8 and divide by 36.

Understanding Sums of Two Dice

This question focuses on finding the probability that the sum of two independently rolled fair dice equals a specific total—in this case, 8. This is a classic higher-tier GCSE problem because it requires counting combinations rather than simply multiplying probabilities. Unlike single-number questions such as “rolling a 6 then a 4,” sums involve recognising that several different ordered pairs can produce the same total. Students often underestimate the importance of order, but each roll of a die is an independent event, and the pair (3,5) is different from (5,3) unless explicitly stated otherwise.

With two six-sided dice, the total number of possible outcomes is 36 because there are 6 choices for the first die and 6 choices for the second, giving a 6×6 grid of ordered pairs. Each of these pairs is equally likely. This structure forms the basis for understanding sums, combinations, and probabilities in multi-step random events.

How to Find All Combinations That Sum to 8

To solve the problem, list all pairs of numbers between 1 and 6 that add up to 8. The pairs are:

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

There are 5 such combinations. Since each pair is equally likely and there are 36 total possible outcomes, the probability becomes \(5/36\). This makes it slightly more likely than many other sums, such as 2 or 12, which have only one combination each. Understanding why some sums appear more frequently is an essential part of deeper probability reasoning.

Step-by-Step Method

  1. Calculate the total number of outcomes: 6 × 6 = 36.
  2. List all pairs that produce a sum of 8.
  3. Count these pairs: there are 5.
  4. Divide the number of favourable outcomes by the total: \(5/36\).

Worked Example 1

Find the probability of rolling a sum of 4. The pairs are (1,3), (2,2), and (3,1), giving 3 favourable outcomes. Probability: \(3/36 = 1/12\).

Worked Example 2

Find the probability of rolling a sum of 10. The valid pairs are (4,6), (5,5), and (6,4). This gives 3 favourable outcomes, so the probability is also \(3/36 = 1/12\).

Common Mistakes

  • Forgetting that (3,5) and (5,3) are different outcomes.
  • Incorrectly counting the total number of possible outcomes—it is always 36 unless the dice are non-standard.
  • Leaving out pairs like (4,4), assuming matching dice outcomes are rare or treated differently.
  • Trying to multiply individual probabilities instead of listing combinations.

Real-Life Applications

Summing dice is a foundation for probability models used in board games, statistical simulations, gaming theory, computer algorithms, and even financial modelling where outcomes depend on combined random events. Recognising that some totals appear more frequently than others helps build intuition about probability distributions.

FAQ

Q: Why is the probability not 1/6?
Because there are multiple ways to make some sums and fewer ways to make others.

Q: Do order and sequence matter?
Yes. (3,5) and (5,3) are separate outcomes because each roll is independent.

Q: Why 36 total outcomes?
6 choices for die 1 × 6 choices for die 2.

Study Tip

When dealing with sums of dice, always write out all pairs. This avoids missed cases and makes probability calculations far more reliable.

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