GCSE Maths Practice: probability-basics

Understanding Impossible Outcomes on a Die

Probability questions involving dice are very common in GCSE Maths. A standard six-sided die is one of the easiest models of randomness because its outcomes are fixed, clearly defined and equally likely. The faces are numbered from 1 to 6. Because of this fixed structure, any number outside this set cannot occur when the die is rolled. Identifying whether an outcome is possible or impossible is an essential skill in probability and is often used in introductory exam questions.

The Sample Space of a Standard Die

The sample space is the full list of all possible outcomes. For a 6-sided die, the sample space is:

{1, 2, 3, 4, 5, 6}

If a number does not appear in this list, then it is impossible to roll that number. This idea underpins the concept of probability-zero events.

Analysing the Answer Choices

5: This is on the die, so rolling a 5 is possible.

6: This is also on the die and is possible to roll.

7: Since the die only has numbers 1–6, the number 7 is not a valid outcome. Therefore, rolling a 7 is an impossible event.

GKSE Maths commonly includes questions like this to reinforce the relationship between the sample space and the probability of events.

The Nature of Impossible Events

An impossible event is one with a probability of 0. This means it cannot occur given the rules of the scenario. Impossible events are different from unlikely events. An unlikely event has a small probability but is still possible. For example, rolling a 6 three times in a row is very unlikely but not impossible. In contrast, rolling a 7 on a six-sided die is completely impossible.

Worked Example 1: Impossible Total on Two Dice

If two fair dice are rolled together, the smallest possible total is 2 (1 + 1), and the largest possible total is 12 (6 + 6). Therefore, a total of 1 or a total of 13 is impossible. These totals fall outside the sample space for adding two dice.

Worked Example 2: Impossible Spinner Outcome

If a spinner has four sections labelled A, B, C and D, then landing on E is impossible because E does not appear anywhere on the spinner. This helps reinforce how the sample space determines what is possible.

Worked Example 3: Impossible Card Draw

In a standard 52-card deck, there is no card called the “15 of hearts.” Since it does not exist in the deck, drawing it is impossible. This mirrors the die example and highlights how probability-zero events depend entirely on the structure of the system.

Common Mistakes

• Confusing impossible with unlikely—unlikely events still have a probability greater than 0.
• Not checking the sample space before deciding if an event is impossible.
• Assuming a die can produce any number, rather than only the numbers printed on its faces.

Real-Life Applications

Understanding impossible events helps in decision-making, risk analysis and data validation. When building probability models, any value outside the sample space should be excluded immediately. This is particularly useful in fields like computer science, statistics and game design, where probability models depend on clear definitions of possible outcomes.

FAQ

Q: Can probability 0 ever become possible?
No. If something is impossible based on the rules or structure, its probability remains 0 unless the rules change.

Q: Could a trick die make rolling a 7 possible?
Only if the die is designed differently. GCSE Maths always assumes a standard six-sided die.

Q: Why is this skill important?
It lays the foundation for evaluating events in more complex scenarios involving multiple probability steps.

Study Tip

Always begin a probability question by identifying the sample space. Once you know all possible outcomes, it becomes easy to see which events are certain, possible, unlikely or impossible.