GCSE Maths Practice: listing-outcomes

Question 2 of 10

Practise counting pair combinations to calculate probabilities for two-dice sums.

‹ ›

\( \begin{array}{l}\textbf{What is the probability of rolling a sum} \\ \textbf{of 8 or 9 with two fair dice?}\end{array} \)

Choose one option:

Break the problem into sums and count combinations carefully.

Understanding Probability of Two-Dice Sums

This question assesses your ability to analyse the full sample space created by rolling two fair six-sided dice. In Higher GCSE probability, students must understand that when two dice are rolled, the outcomes form ordered pairs. Since each die has 6 outcomes, the total number of possible results is 36. These outcomes are equiprobable, meaning each has the same chance of occurring.

When working with sums from two dice, the key is recognising how many combinations produce each possible total. Some sums, such as 7, have many combinations. Others, like 2 or 12, have only one. This structure creates different likelihoods for different totals.

Step-by-Step Breakdown

  1. Identify the event of interest: rolling a total of 8 or 9.
  2. List all pairs that sum to 8.
  3. List all pairs that sum to 9.
  4. Add the counts together to get the total favourable outcomes.
  5. Divide by the total number of outcomes (36) to obtain the probability.

Pairs That Sum to 8

The combinations that produce 8 are:

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

That gives 5 favourable outcomes.

Pairs That Sum to 9

The combinations are:

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

That gives 4 favourable outcomes.

Adding them yields a total of 9 favourable combinations.

Total Probability

Since there are 36 possible results when rolling two dice:

Probability = 9 / 36 = 1 / 4

Although 1/4 is the simplified version, the form 9/36 is perfectly acceptable, especially when focusing on counting methods.

Worked Example 1: Finding the Probability of a Sum of 6

Sums to 6: (1,5),(2,4),(3,3),(4,2),(5,1) → 5 outcomes. Probability = 5/36.

Worked Example 2: Probability of Doubles

Doubles: (1,1),(2,2),(3,3),(4,4),(5,5),(6,6) → 6 outcomes. Probability = 6/36 = 1/6.

Worked Example 3: Probability of Getting a Sum Greater Than 9

Sums 10, 11, 12 give 6 favourable outcomes, so Probability = 6/36 = 1/6.

Common Mistakes

  • Adding instead of multiplying 6×6. Some students incorrectly think there are only 12 outcomes.
  • Forgetting that (3,5) and (5,3) are different. Order matters with two dice.
  • Missing pairs or double-counting pairs. Listing carefully prevents this.
  • Confusing sums with products. The total refers only to addition.

Real-Life Applications

Dice-based probability is used in board games, gaming algorithms, computer simulations, risk modelling, and statistical teaching tools. Understanding how sums form helps students recognise patterns and build confidence for more challenging probability tasks such as distributions and expected values.

FAQ

Q: Why 36 outcomes?
A: Each die has 6 possibilities, so 6 × 6 = 36 ordered outcomes.

Q: Why do we count ordered pairs separately?
A: (3,6) and (6,3) represent different physical outcomes.

Q: Which sums are most common?
A: 6, 7, and 8 because they have the highest number of combinations.

Study Tip

When solving two-dice problems, draw a 6×6 grid. It helps ensure that every combination is counted once and only once.

Related Quizzes