GCSE Maths Practice: listing-outcomes

Understanding Sums of Two Dice in Higher GCSE Probability

This question requires analysing the full outcome space of rolling two six-sided dice and identifying which combinations produce a total of 10 or more. Because it involves recognising patterns in sums, listing combinations, and dividing by the total number of outcomes, it fits perfectly into Higher-level GCSE probability. The key idea is that each die roll is independent, meaning the total number of outcomes is found using multiplication: 6 outcomes on the first die multiplied by 6 outcomes on the second die gives 36 possible ordered pairs.

Working with sums of two dice is a common probability task. Some sums occur more frequently than others because more combinations lead to them. For example, a sum of 7 has six combinations, making it the most likely total. In contrast, extreme values like 2 or 12 only have one combination each.

Step-by-Step Breakdown

1. Write out all possible outcomes: there are 36 ordered pairs.
2. List all combinations that give a sum of 10, 11, or 12.
3. Count the number of favourable outcomes.
4. Divide this number by 36 to form the probability.
5. Simplify the resulting fraction if needed.

This method ensures accuracy and helps students build strong habits for more complex questions later.

All Combinations Giving Sums ≥ 10

Sum of 10
(4,6), (5,5), (6,4)
These three combinations show that 10 can be formed three different ways.

Sum of 11
(5,6), (6,5)
Only two combinations produce 11.

Sum of 12
(6,6)
This is the only way to make 12.

Adding them together: 3 + 2 + 1 = 6 favourable outcomes.

Why the Probability Is 1/6

Since there are 36 possible outcomes and 6 of them meet the condition, the probability is:

6 ÷ 36 = 1 ÷ 6 = 1/6

This shows that sums of 10 or more appear in about 16.7% of two-dice rolls.

Worked Example 1: Probability of Rolling a Sum of 8

Combinations: (2,6), (3,5), (4,4), (5,3), (6,2). There are 5 favourable outcomes. Probability = 5/36.

Worked Example 2: Probability of Rolling Less Than 5

Totals 2, 3, and 4. This gives combinations: 2: (1,1); 3: (1,2),(2,1); 4: (1,3),(2,2),(3,1). Total favourable = 6. Probability = 6/36 = 1/6.

Worked Example 3: Probability of Getting Doubles

Doubles include (1,1), (2,2), ..., (6,6). That makes 6 favourable outcomes. Probability = 6/36 = 1/6. Notice this matches the same probability as rolling a sum ≥ 10.

Common Mistakes

• Using 12 outcomes instead of 36. This happens when learners incorrectly add 6 + 6 instead of multiplying.
• Missing reversed pairs. For example, (4,6) and (6,4) are both different outcomes.
• Forgetting to include all sums. Some students list only 10 and 11 and forget 12.
• Confusing sums with products or differences. Only totals matter here.

Real-Life Applications

This probability scenario appears in board games, simulations, and decision-making models. Understanding dice sums helps with analysing fairness in games, designing simulations, and interpreting random processes. Many higher-level mathematical and statistical concepts build upon these foundational probability skills.

FAQ

Q: Why do we treat (4,6) and (6,4) separately?
A: Because dice have an order — the first and second roll are independent events.

Q: Can the probability be left as 6/36?
A: Yes, but simplified forms are preferred.

Q: Which sums are least likely?
A: 2 and 12 — each has only one combination.

Study Tip

Create a 6×6 outcome grid whenever solving two-dice problems. It prevents missing combinations and helps you see patterns clearly.