Probability Basics and Relative Frequency –

Probability is one of the most important areas of GCSE Maths and features heavily on AQA, Edexcel and OCR papers at both Foundation and Higher level. You need to understand how to calculate theoretical probabilities, use the probability scale, list outcomes systematically, and apply relative frequency to estimate probabilities from experiments. These skills are the building blocks for more advanced topics such as tree diagrams, Venn diagrams and conditional probability. This guide explains every concept clearly, provides worked examples at both tiers, warns you about the mistakes examiners see most often, and gives you questions to practise. For an overview of every topic, see our complete GCSE Maths topics list.

What Is Probability?

Probability measures how likely an event is to happen. It is always a value between 0 and 1 (or equivalently between 0% and 100%).

• A probability of 0 means the event is impossible.
• A probability of 1 means the event is certain.
• A probability of 0.5 means the event is equally likely to happen or not happen.

Theoretical Probability

When all outcomes are equally likely, you calculate probability using:

For example, the probability of rolling a 3 on a fair six-sided dice is 1/6.

Relative Frequency (Experimental Probability)

When outcomes are not equally likely — or you do not know the theoretical probability — you estimate it using relative frequency:

The more trials you carry out, the more reliable the estimate becomes. As the number of trials increases, relative frequency tends towards the theoretical probability.

The Probability Scale

Probabilities can be expressed as fractions, decimals or percentages. They can be placed on a probability scale from 0 to 1:

• 0 — impossible
• 0.25 — unlikely
• 0.5 — even chance
• 0.75 — likely
• 1 — certain

Complementary Events

The probability that an event does not happen is:

For example, if P(rain) = 0.3, then P(no rain) = 1 − 0.3 = 0.7.

Exhaustive and Mutually Exclusive Events

• Exhaustive events cover all possible outcomes. Their probabilities add up to 1.
• Mutually exclusive events cannot happen at the same time. If A and B are mutually exclusive: P(A or B) = P(A) + P(B).

Step-by-Step Method

Calculating Theoretical Probability

1. List all possible outcomes (the sample space).
2. Count the total number of outcomes.
3. Count how many outcomes satisfy the event.
4. Divide: favourable ÷ total.
5. Simplify the fraction if possible.

Using Relative Frequency

1. Carry out or read the results of an experiment.
2. Count how many times the event occurred.
3. Divide by the total number of trials.
4. State that this is an estimate — it becomes more reliable with more trials.

Expected Frequency

You can predict how many times an event should occur:

Worked Example 1 — Foundation Level

Question: A bag contains 3 red, 5 blue and 2 green marbles. A marble is picked at random. (a) Find the probability of picking a blue marble. (b) Find the probability of not picking a green marble.

Working:

Total marbles = 3 + 5 + 2 = 10.

(a) P(blue) = 5/10 = 1/2.

(b) P(green) = 2/10 = 1/5. P(not green) = 1 − 1/5 = 4/5.

Worked Example 2 — Higher Level

Question: A spinner is spun 200 times. It lands on red 68 times. (a) Calculate the relative frequency of landing on red. (b) The spinner is spun 500 more times. Estimate how many times it will land on red. (c) Do you think the spinner is fair if it has 4 equal sections? Explain.

Working:

(a) Relative frequency = 68 ÷ 200 = 0.34.

(b) Expected frequency = 0.34 × 500 = 170 times.

(c) If the spinner were fair with 4 equal sections, the theoretical probability of red would be 1/4 = 0.25. The relative frequency of 0.34 is noticeably higher than 0.25. With 200 trials this is a reasonable sample size, so the spinner may not be fair — it appears biased towards red. However, more trials would increase reliability.

Common Mistakes

• Giving a probability greater than 1 or less than 0 — if your answer is outside 0 to 1, you have made an error.
• Confusing theoretical and experimental probability — theoretical probability uses equally likely outcomes; relative frequency uses experimental results. Do not mix them up.
• Forgetting to simplify — always simplify fractions where possible.
• Saying "1 in 6" instead of 1/6 — probabilities should be expressed as fractions, decimals or percentages, not as ratios or "1 in…" statements.
• Assuming the complement is 0.5 — P(not A) = 1 − P(A), not automatically 0.5.
• Small sample conclusions — do not draw strong conclusions from a small number of trials. State that more trials are needed for reliability.

Exam Tips

1. Always give probabilities as fractions, decimals or percentages unless the question specifies otherwise.
2. Use the word "estimate" when working with relative frequency — this shows the examiner you understand the difference between theoretical and experimental probability.
3. List outcomes systematically — use tables, lists or sample space diagrams to avoid missing outcomes. For two dice, a two-way table ensures you list all 36 outcomes.
4. Check that probabilities sum to 1 — if the question gives you probabilities for all but one outcome, use 1 minus the sum to find the missing probability.
5. Expected frequency questions often follow relative frequency — multiply the probability by the number of trials.
6. For tree diagram extensions, see probability tree diagrams. For key formulas, visit our GCSE Maths formulas page.

Practice Questions

Question 1 (Foundation): A fair six-sided dice is rolled. Find the probability of rolling (a) a 4, (b) an even number, (c) a number greater than 4.

Question 2 (Foundation): The probability of picking a yellow sweet from a bag is 0.35. What is the probability of not picking a yellow sweet?

Question 3 (Higher): A coin is flipped 80 times and lands on heads 52 times. (a) Calculate the relative frequency of heads. (b) Is the coin fair? Explain your answer.

Question 4 (Higher): The probability of a bus being late is 0.15. In 60 journeys, how many times would you expect the bus to be late?

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Related Topics

• Probability Tree Diagrams
• Venn Diagrams
• Conditional Probability
• Sampling Methods

Summary

Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain). Theoretical probability divides favourable outcomes by total equally likely outcomes. Relative frequency estimates probability from experimental results — the more trials, the more reliable the estimate. Use P(not A) = 1 − P(A) for complementary events, and P(A or B) = P(A) + P(B) for mutually exclusive events. Expected frequency multiplies probability by the number of trials. Always express probabilities as fractions, decimals or percentages, simplify where possible, and remember that relative frequency is an estimate, not an exact value. These fundamentals underpin tree diagrams, Venn diagrams and conditional probability at Higher level.