Experimental Probability, Relative Frequency and Sample Size Revision Notes | GCSE AQA Higher Maths
The law of large numbers means that as the number of trials increases, the relative frequency gets closer to the theoretical probability. Free notes.
Updated 24 Mar 2026 · Revise Right Team
Experimental Probability, Relative Frequency and Sample Size is part of Probability Foundations in GCSE AQA Higher Maths.
4 min
Reading time
12
Lessons in this topic
GCSE AQA Higher
Pathway
Reviewed by Revise Right
Content review
What this lesson covers
The law of large numbers means that as the number of trials increases, the relative frequency gets closer to the theoretical probability. Free notes.
Calculate relative frequency from experimental data and express it as a fraction, decimal, or percentage
Explain why relative frequency is used as an estimate of probability when theoretical probability is unknown
Compare experimental and theoretical probability for a given scenario
Describe how increasing sample size causes relative frequency to converge towards theoretical probability
Revise Right Mocks '26
AQA GCSE Maths mock draft, 26–31 Oct. Draft coverage: Maths Paper 1 (Higher tier). Reviewed for an original-practice sitting; check student availability before registering. Marking and results will follow later, after separate approval. Eligible new accounts receive a five-day Pro trial when signup email verification completes. Previous trials keep their original expiry. After expiry, new papers need Pro unless you have an explicitly confirmed free sitting offer for this event. That offer adds no Pro time or AI learning. Saved attempts and released results remain readable. No card or automatic charge.
Use this lesson to stay inside Probability Foundations while connecting the detail back to the wider Probability topic.
Relative Frequency: A probability estimate calculated from experimental data. Found by dividing the number of times an outcome occurs by the total number of trials. Used when theoretical probability cannot be calculated directly.
Experimental Probability: A probability value obtained by carrying out an experiment or observing real events. It equals the relative frequency of the outcome across all trials conducted.
Theoretical Probability: A probability value calculated from known equally likely outcomes, without needing to run an experiment. Used when all outcomes are known and equally likely.
Sample Size: The total number of trials or observations in an experiment. Larger sample sizes produce relative frequencies that are closer to the true theoretical probability.
Revision notes
Read the core explanations from Experimental Probability, Relative Frequency and Sample Size before testing yourself from memory.
1. Why Experimental Probability Matters
You already know that a fair coin has a theoretical probability of $\frac{1}{2}$ for heads. But what if someone hands you a coin and asks: "Is this coin fair?" You cannot answer that from theory alone — you have to test it.
That is exactly where experimental probability comes in. Instead of reasoning from equally likely outcomes, you run trials and record what actually happens. The proportion of times an outcome occurs is called its relative frequency, and it acts as your best estimate of the true probability.
This idea sits at the heart of the specification requirement you need to master: understanding that as sample size grows, relative frequency tends towards the theoretical probability — provided the experiment is unbiased.
2. Calculating Relative Frequency
The formula is straightforward, but the skill lies in applying it carefully and reading the question correctly. Here is the core formula you will use every time.
Notice that relative frequency is always between 0 and 1 (inclusive). A relative frequency of 0 means the outcome never occurred; a relative frequency of 1 means it occurred every single time. This mirrors the probability scale you already know.
Worked example: Calculating Relative Frequency from a Frequency Table
Mohammed rolls a six-sided die 200 times. The number 4 appears 38 times. Calculate the relative frequency of rolling a 4, and compare it to the theoretical probability.
Number of times 4 appeared $= 38$
Total number of rolls $= 200$ — Read these directly from the question. [M1] for correct identification.
$$\text{Relative Frequency} = \frac{38}{200}$$ — Substitute into the formula. [M1] for correct substitution.
$$\frac{38}{200} = \frac{19}{100} = 0.19$$ — Divide numerator and denominator by 2. [A1] for correct simplified value.
$$P(\text{rolling a 4}) = \frac{1}{6} \approx 0.167$$ — A fair die has 6 equally likely outcomes, so theoretical probability is $\frac{1}{6}$.
Relative frequency $= 0.19$
Theoretical probability $\approx 0.167$
Difference $= 0.19 - 0.167 = 0.023$ — The relative frequency is close to but not equal to the theoretical probability. With more trials, we would expect them to get even closer.
Answer: Relative frequency $= 0.19$ (or $\frac{19}{100}$). This is close to the theoretical probability of $\frac{1}{6} \approx 0.167$, with a small difference due to random variation in 200 trials.
3. Sample Size and Convergence to Theoretical Probability
Here is the central idea of this topic: the more trials you run, the closer your relative frequency gets to the true theoretical probability — as long as the experiment is unbiased. This is the key Higher-tier concept you must be able to explain clearly.
Think about flipping a fair coin. After 10 flips, you might get 7 heads — a relative frequency of 0.7, which looks far from 0.5. After 1000 flips, you are very likely to get something close to 500 heads, giving a relative frequency near 0.5. The randomness does not disappear, but its effect on the overall proportion shrinks as the sample grows.
The graph above shows a typical pattern: early results jump around unpredictably, but the line gradually flattens and homes in on the theoretical value. Examiners often ask you to describe or explain this pattern — so the language you use matters.
Worked example: Describing the Effect of Sample Size on Relative Frequency
Ben conducts an experiment rolling a fair die and records the relative frequency of rolling a 6 after different numbers of trials. His results are shown below. Explain what the data shows about sample size and relative frequency.
Trials: 10, 50, 100, 500, 1000
Relative frequency of 6: 0.40, 0.28, 0.22, 0.18, 0.17
$$P(\text{rolling a 6}) = \frac{1}{6} \approx 0.167$$ — Establish the target value the relative frequency should approach.
After 10 trials: relative frequency $= 0.40$
Difference from $\frac{1}{6}$: $0.40 - 0.167 = 0.233$ — Small sample sizes produce large deviations from the theoretical probability due to random variation.
After 1000 trials: relative frequency $= 0.17$
Difference from $\frac{1}{6}$: $0.17 - 0.167 = 0.003$ — With a large sample, the relative frequency is very close to the theoretical probability.
As the number of trials increases, the relative frequency of rolling a 6 gets closer to the theoretical probability of $\frac{1}{6} \approx 0.167$. With small samples, results vary widely; with large samples, the relative frequency converges towards the theoretical value. — This is the key explanation the examiner is looking for. Use the words 'increases', 'closer to', and 'theoretical probability'.
Answer: As sample size increases, the relative frequency converges towards the theoretical probability of $\frac{1}{6}$. Small samples show large random variation; large samples produce stable estimates close to the true probability.
4. Unbiased Samples and When Relative Frequency is Reliable
The convergence described above only holds when the experiment is unbiased. An unbiased experiment is one where every trial is independent and the conditions do not favour any particular outcome artificially.
If a spinner is weighted on one side, or if a survey only asks people who already agree with a viewpoint, the relative frequency will converge — but towards the wrong value. It will home in on the biased probability, not the fair theoretical one. This is why the specification links sample size to unbiased samples specifically.
Each trial must be independent — the result of one trial must not affect the next.
The experiment must be conducted under consistent conditions throughout.
The sample must be representative of the population being studied.
A larger unbiased sample gives a more reliable estimate of theoretical probability.
5. Worked Practice and Exam Application
Now that the core ideas are in place, it is time to apply them to the kinds of questions you will meet in the exam. Examiners test this topic in two main ways: calculation questions (find the relative frequency) and explanation questions (describe what happens as sample size changes).
Worked example: Multi-Part Exam Question on Relative Frequency
Chloe tests a biased coin by flipping it repeatedly. Her results are recorded in the table below.
Number of flips: 20, 100, 500
Number of heads: 14, 62, 295
(a) Calculate the relative frequency of heads after each set of flips.
(b) Estimate the probability of getting heads with this coin. Justify your answer.
Relative frequencies: 0.70, 0.62, 0.59
The values are decreasing and appear to be settling. — As sample size increases, the relative frequency becomes more stable — this is convergence in action.
Best estimate: $P(\text{heads}) \approx 0.59$
Justification: The largest sample (500 flips) gives the most reliable estimate because relative frequency converges to the true probability as sample size increases. — [A1] for value close to 0.59 with correct justification referencing sample size. Always use the largest sample for your best estimate.
Answer: (a) After 20 flips: 0.70; after 100 flips: 0.62; after 500 flips: 0.59. (b) Best estimate is approximately 0.59, using the 500-flip result as it has the largest sample size and therefore gives the most reliable estimate.
6. Exam Technique and Common Mistakes
This topic rewards precise language. Calculation questions are usually straightforward, but explanation questions are where marks are lost. The examiner wants to see specific vocabulary: increases, converges, tends towards, theoretical probability, and sample size.
The core of this topic is a single, powerful idea: run more trials, get a better estimate. Relative frequency is your experimental tool for estimating probability, and its reliability is directly tied to how many trials you conduct in an unbiased experiment — exactly what the specification requires you to understand and explain.
Sources and content standards
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-24.
At GCSE level, the law of large numbers means that as the number of trials increases, the relative frequency gets closer to the theoretical probability. The randomness does not disappear, but its effect on the overall proportion shrinks as the sample grows.
What happens to relative frequency as the number of trials increases?
As the number of trials in an unbiased experiment increases, the relative frequency tends towards the theoretical probability. With a fair die, the relative frequency of a 6 might be 0.40 after 10 trials but 0.17 after 1000, close to 1/6 ≈ 0.167.
How do you find the best estimate of a probability from experiments?
Use the relative frequency from the largest sample, because it gives the most reliable estimate. If a biased coin gives 14 heads in 20 flips, 62 in 100 and 295 in 500, the best estimate is 295/500 = 0.59.
What makes an experiment unbiased?
An unbiased experiment has independent trials, consistent conditions and a representative sample. If a spinner is weighted, the relative frequency still converges, but towards the biased probability rather than the fair theoretical one.
Why is relative frequency used as an estimate of probability?
Relative frequency is used when theoretical probability is unknown, for example to test whether a coin is fair. If a die shows 4 on 38 of 200 rolls, the relative frequency is 38/200 = 0.19, close to the theoretical probability of 1/6 ≈ 0.167.
What is the relative frequency if a 4 is rolled 38 times in 200 rolls?
The relative frequency is 38/200 = 19/100 = 0.19. This is close to the theoretical probability of 1/6 ≈ 0.167, with a small difference of about 0.023 due to random variation in 200 trials.
Why are small samples unreliable in probability experiments?
Small samples are not reliable because random variation has a big effect on the overall proportion. After 10 rolls the relative frequency of a 6 was 0.40, far from 1/6 ≈ 0.167, but after 1000 rolls it was 0.17.
What does converge mean in probability?
Converge means get closer and closer to a value. As the number of trials in an unbiased experiment increases, the relative frequency of an outcome converges towards, or tends towards, its theoretical probability.
Does relative frequency converge for a biased spinner?
Relative frequency still converges for a biased spinner, but towards the biased probability rather than the fair theoretical one. This is why convergence to the theoretical probability needs an unbiased experiment.
Can relative frequency be written as a percentage?
Relative frequency can be given as a fraction, decimal or percentage. For example, 38 out of 200 is 19/100, 0.19 or 19%, and it always lies between 0 and 1 as a decimal.
How do you explain the effect of sample size on relative frequency?
The effect of sample size is explained by stating that as the number of trials increases, the relative frequency gets closer to the theoretical probability. Answers should use words like increases, converges, tends towards, theoretical probability and sample size.