GCSE Maths Practice: listing-outcomes

Understanding Possible Outcomes When Rolling Two Dice

This question focuses on identifying which pairs of numbers can occur when rolling two standard six-sided dice. Each die in a standard set has the numbers 1 through 6 printed on its faces. Because both dice are independent and identical in structure, any combination of two numbers between 1 and 6 is a valid outcome. Understanding this idea helps build a strong foundation for probability topics such as sample spaces, tree diagrams, frequency tables, and combined-event probability.

When rolling two dice, the complete sample space consists of 36 possible outcomes. This is because each die has 6 numbers, and the total combinations are found using the multiplication rule: 6 × 6 = 36. Each outcome is written as a pair, such as (3, 5), where the first number represents the result of the first die and the second number represents the result of the second die. Since each die functions independently, every combination has the same chance of occurring.

Checking Whether a Pair Is a Valid Outcome

To decide if a pair is possible, both numbers must fall within the range 1 to 6. Any value outside this range cannot appear on a standard die. For example, a result like 0, 8, or 10 would be impossible, as well as any negative numbers. Valid pairs include all combinations where both values are within the allowed range.

Step-by-Step Method

1. Recall that each die produces numbers from 1 to 6.
2. Look at each pair individually.
3. Check both numbers in the pair.
4. If both numbers are between 1 and 6, the pair is a valid outcome.
5. If either number falls outside this range, the pair is impossible.

This simple checking method ensures accuracy in identifying possible outcomes.

Worked Example 1: Checking the Pair (3, 4)

Both 3 and 4 lie between 1 and 6. Therefore, (3, 4) is a valid outcome when rolling two dice. This combination is part of the 36-element sample space.

Worked Example 2: Checking the Pair (6, 9)

The number 9 is not a possible result on a six-sided die. Therefore, the pair (6, 9) cannot occur. Any number above 6 or below 1 automatically makes the pair invalid.

Worked Example 3: Checking the Pair (0, 5)

The number 0 does not appear on any face of a standard die. Although 5 is valid, 0 is not, making the pair (0, 5) an impossible outcome.

Common Mistakes

• Assuming dice can show numbers higher than 6. Some students incorrectly believe dice include numbers like 7 or 8.
• Treating the order as irrelevant. (2,5) and (5,2) are different outcomes because they represent different dice.
• Mixing up the sample space. Forgetting that there are 36 total outcomes can lead to confusion in later probability calculations.
• Thinking repeated numbers are impossible. Pairs like (1,1) or (4,4) are perfectly valid.

Real-Life Applications

Dice outcomes are used in board games, probability experiments, statistical modelling, and simulations. Understanding valid outcomes helps students interpret patterns, calculate theoretical and experimental probabilities, and analyse fair games. In computing and game design, virtual dice operate on the same principles and rely on uniformly distributed random numbers.

FAQ

Q: Are pairs like (1,1) allowed?
A: Yes. Rolling the same number on both dice is common and completely valid.

Q: Can dice show a 7?
A: No. Standard dice only include 1–6.

Q: Is (2,5) the same as (5,2)?
A: No. They represent different orders of results.

Study Tip

When unsure whether a pair is possible, quickly refer to the allowed range: 1 to 6. If both numbers fit within that range, the pair is a valid outcome.