GCSE Maths Practice: listing-outcomes

Question 4 of 10

Practise working with outcome spaces to calculate two-dice probabilities.

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\( \begin{array}{l}\textbf{What is the probability of getting a sum of 7} \\ \textbf{when rolling two fair 6-sided dice?}\end{array} \)

Choose one option:

Remember that dice outcomes are ordered pairs, not sums only.

Understanding Probability of Sums with Two Dice

This question tests an essential Higher GCSE probability skill: analysing all possible outcomes from two independent events and identifying how many of those outcomes satisfy a given condition. In this case, you are rolling two fair six-sided dice and calculating the probability that their total is exactly 7. Because dice problems frequently appear in probability exams, understanding how to construct and interpret the outcome space is extremely important.

A single six-sided die has six possible outcomes. When rolling two dice together, the total outcomes are not 6 + 6, but 6 × 6 = 36. This is because each number on the first die can pair with each number on the second die. These outcomes are usually represented as ordered pairs, such as (2, 5), where the first number represents the result of the first die and the second number the result of the second die.

Why the Sum of 7 Is Special

The total of 7 is one of the most common sums rolled with two dice. Because there are multiple different combinations that all lead to the same sum, the probability is higher than many other totals. To find the probability, we list every combination that produces a sum of 7. The valid combinations are:

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

These 6 outcomes are considered favourable because they meet the condition of giving a total of 7.

Step-by-Step Breakdown

  1. Write the total number of outcomes: 36.
  2. List all combinations that sum to 7.
  3. Count those combinations — there are 6.
  4. Form the probability: 6 ÷ 36.
  5. Reduce the fraction: 1/6.

This structured method works for any sum or condition involving two dice.

Worked Example 1: Probability of Getting a Sum of 5

Pairs that sum to 5 are (1, 4), (2, 3), (3, 2), and (4, 1). That gives 4 favourable outcomes. The probability is 4/36 = 1/9.

Worked Example 2: Probability of Getting a Sum of 11

Pairs that sum to 11 are (5, 6) and (6, 5). That gives 2 favourable outcomes. Probability = 2/36 = 1/18.

Worked Example 3: Probability of Getting Doubles

Doubles include (1,1), (2,2), ..., (6,6). There are 6 favourable outcomes. Probability = 6/36 = 1/6, the same as the probability of getting a sum of 7.

Common Mistakes

  • Adding instead of multiplying outcomes. Students sometimes think there are 12 outcomes instead of 36.
  • Missing ordered pairs. (3,4) and (4,3) are different outcomes and must both be counted.
  • Listing incomplete combinations. Leaving out a pair changes the entire probability calculation.
  • Thinking some pairs occur more often. All 36 combinations are equally likely.

Real-Life Applications

Dice-based probability is used in board games, simulations, statistics, casino game design, and computer modelling. Understanding the structure of outcome spaces helps students develop strong reasoning skills applicable to probability trees, experimental probability, and theoretical modelling.

FAQ

Q: Are (3,4) and (4,3) the same?
A: No. They represent different orders and are separate outcomes.

Q: Why is the total number of outcomes 36?
A: Because each die has 6 outcomes and the events are independent, giving 6 × 6 combinations.

Q: Do sums near the middle occur more often?
A: Yes. Totals like 6, 7, and 8 have more combinations than extreme totals like 2 or 12.

Study Tip

When working with two-dice problems, always construct the full outcome table or list pair combinations. This prevents missing outcomes and ensures complete accuracy.

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