GCSE Maths Practice: probability-basics

Understanding the Fundamental Rules of Probability

Probability theory begins with a few essential rules that describe how likely an event is to occur. These rules apply to every probability question in GCSE Maths, from simple dice problems to combined events, Venn diagrams and conditional probability. A deep understanding of these principles helps students avoid common mistakes and build strong reasoning skills for higher-tier questions.

Probability Values Must Be Between 0 and 1

Every probability is expressed as a number between 0 and 1 inclusive. A value of 0 means an event cannot happen, and a value of 1 means an event is certain. Probabilities may be written as fractions, decimals or percentages. For example, 1/2, 0.5 and 50% all represent the same probability. No matter which format is used, the value must always lie within the correct range.

Why Probabilities Cannot Exceed 1

A probability greater than 1 would imply that an event is more than certain, which is impossible. For example, a statement such as “the probability is 1.2” is invalid because probabilities measure proportions of possible outcomes. A proportion cannot logically exceed the total of all possibilities.

Impossible Events Have Probability 0

An impossible event is one that cannot occur under any circumstances. For example, rolling a 7 on a standard six-sided die is impossible. Its probability is 0. This differs from unlikely events, which have very small probabilities but can still occur. For example, rolling six 6s in a row is extremely unlikely, yet its probability is still greater than zero.

Worked Example 1: Valid or Invalid Probabilities

If a question states that the probability of an event is −0.3, this is invalid because probabilities cannot be negative. Similarly, a value such as 1.4 is invalid because probabilities cannot exceed 1. Identifying invalid probability statements is often assessed in GCSE reasoning questions.

Worked Example 2: Real-World Probability Interpretation

Suppose weather data predicts that the chance of rain tomorrow is 0.8. This means there is an 80% chance it will rain. The value is valid because it lies between 0 and 1. A value of 1 would indicate guaranteed rain, while a value of 0 would indicate rain is impossible.

Worked Example 3: Impossible and Certain Events

If you pick a number from the set {1, 2, 3}, the probability that the number is 4 is 0, because 4 is not in the set. On the other hand, the probability that the number is less than 10 is 1, because every number in the set satisfies the condition.

Common Mistakes

• Thinking probability values can exceed 1 or be negative.
• Confusing unlikely events with impossible events.
• Thinking an impossible event has probability 1 instead of 0.
• Mixing up probability with frequency—probability describes the chance, not the actual outcome.

Real-Life Applications

Probability rules are used in risk assessment, finance, science experiments and medical decision-making. Ensuring that values fall between 0 and 1 is essential for accurate statistical modelling. Understanding impossible and certain events also supports logical thinking, which is valuable in subjects such as computer science and data science.

FAQ

Q: Can a probability be exactly 0?
Yes. This means the event cannot happen.

Q: Can a probability be exactly 1?
Yes. This means the event is guaranteed to happen.

Q: Can probabilities be written as percentages?
Yes, as long as the value is between 0% and 100%.

Study Tip

Whenever you see a probability written as a number, quickly check whether it lies between 0 and 1. This simple habit prevents errors and strengthens your understanding of probability rules across all GCSE topics.