GCSE Maths Practice: probability-basics

Understanding Impossible Events in GCSE Probability

In GCSE Maths, probability questions often require you to identify whether events are certain, likely, unlikely or impossible. An impossible event is one that cannot happen under any circumstances within the rules of the situation. Its probability is always 0. This question builds an essential skill: evaluating whether an outcome is part of the sample space. If an outcome is not included in the sample space, the probability of that event is zero.

The Sample Space

The sample space is the set of all possible outcomes. For example, when rolling a fair six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. Any number outside this list cannot occur. Recognising the boundaries of the sample space makes it much easier to determine which events are impossible.

Analysing Each Option

Rolling a 7 on a fair 6-sided die: The die only has numbers 1 to 6, so 7 is not part of the sample space. This event is impossible.

Flipping a coin and getting both heads and tails: A single coin flip has only one outcome. You can get heads or tails, but not both at the same time. Therefore, this outcome is impossible.

Drawing a 5 of spades from a deck of cards: This is possible because every standard 52-card deck includes the 5 of spades. Since the event can occur, its probability is greater than zero.

Worked Example 1: Impossible Dice Event

If a question asks: “What is the probability of rolling an 11 on two dice?” you must consider the sample space. The highest possible total on two dice is 12 (6 + 6). Since 11 is within the possible totals, this event is not impossible. But rolling a total of 1 on two dice is impossible because the smallest total is 2 (1 + 1). Recognising these limits is crucial.

Worked Example 2: Impossible Card Event

If a question asks for the probability of drawing the 15 of hearts, this is impossible because no card numbered 15 exists. Similarly, asking for the 0 of clubs or the 20 of diamonds would also represent impossible events.

Worked Example 3: Impossible Spinner Event

If a spinner is divided into 4 equal sections labelled A, B, C and D, it is impossible to land on E. Since E does not appear on the spinner, it is outside the sample space and therefore has probability 0.

Common Mistakes

• Confusing rare events with impossible events—rare events still have a probability greater than 0.
• Not checking the sample space before judging whether an outcome is possible.
• Assuming events involving multiple outcomes (e.g., both heads and tails) are possible in one trial, forgetting the idea of a single action.

Real-Life Applications

Understanding impossible events helps with risk assessment and scenario modelling. For example, if a company knows certain outcomes cannot occur, they can eliminate them when analysing data. In scientific modelling, events outside the sample space are disregarded because they cannot physically occur. This concept later supports GCSE and A-level topics involving mutually exclusive events, sets and Venn diagrams.

FAQ

Q: Can an impossible event ever happen?
No. By definition, an impossible event has probability 0 and never occurs.

Q: Can probability 0 and probability close to 0 be the same?
No. A very unlikely event may have a tiny probability, but not zero. Probability 0 means it cannot occur at all.

Q: Do sample spaces always help identify impossible events?
Yes. Anything not included in the sample space is impossible.

Study Tip

When checking if an event has probability 0, list the sample space first. If the event is not in that list, the probability is automatically 0. This method keeps your reasoning structured and reliable during GCSE probability questions.