Relative Frequency and Experimental Probability — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers estimating a probability from results actually observed, rather than working it out from the structure of the situation. By the end of this guide you should be able to calculate a relative frequency and use it to predict how often an event will happen.

You should also be able to compare an experimental result with a theoretical probability, decide whether an object is biased, explain why more trials give a better estimate, and complete a relative frequency table.

The organising idea is that relative frequency is what you do when you cannot reason the probability out. For a fair die you can say the probability of a six is 1/6 without rolling it once, because the six faces are identical. For a drawing pin, nothing about its shape tells you how often it lands point-up — the only way to find out is to throw it many times and count. That distinction runs through every question here: theoretical probability comes from the structure, experimental probability comes from the evidence, and the more evidence you gather the closer the two should come.

Key terms and definitions

Trial — one repetition of an experiment.

Frequency — the number of times an outcome occurred.

Relative frequency — the frequency divided by the number of trials.

Experimental probability — another name for relative frequency, used as an estimate of the true probability.

Theoretical probability — a probability worked out from the structure of the situation, without experimenting.

Biased — not fair, so the outcomes are not equally likely.

Expected frequency — the number of occurrences predicted, found by multiplying the probability by the number of trials.

Core concepts

Calculating relative frequency

Relative frequency = frequency of the event ÷ total number of trials.

If a drawing pin lands point-up 36 times in 100 throws, the relative frequency is 36 ÷ 100 = 0.36.

Like any probability, it always lies between 0 and 1, and it may be written as a fraction, decimal or percentage. A value outside that range means the arithmetic has gone wrong, usually by dividing the wrong way round.

The denominator is the total number of trials, not the number of other outcomes. With 36 point-up out of 100, the denominator is 100, not 64.

When to use it instead of theory

Use the theoretical approach whenever the outcomes are equally likely — a fair coin, a fair die, a well-shuffled pack. The words fair and unbiased are the signal, and no experiment is needed.

Use relative frequency when the outcomes are not equally likely, or when you have no way of knowing: a drawing pin, a bent coin, a weighted spinner, or a real-world event such as whether a bus is late.

A question giving you a table of results is telling you to use relative frequency; a question describing a fair object is telling you not to.

More trials give a better estimate

Relative frequency is an estimate, and its reliability depends almost entirely on the number of trials.

With 10 throws of a fair coin, getting 7 heads is unremarkable, giving a relative frequency of 0.7 — a long way from the true 0.5. With 1000 throws, a result as extreme as 700 heads would be astonishing.

So as the number of trials increases, the relative frequency generally settles down and moves closer to the true probability. That is the statement examiners look for, and it earns the mark where "do more trials" alone does not.

It is worth adding that the improvement is not guaranteed on any particular run — it is a tendency, not a rule, and the relative frequency can wander before settling.

Deciding whether something is biased

Compare the relative frequency with the theoretical probability the object would have if it were fair.

A fair six-sided die should give each number a relative frequency near 1/6, which is about 0.17. If 300 rolls produce 95 sixes, the relative frequency is 0.32 — roughly double what is expected over a large number of trials, which is good evidence of bias.

Two things must be mentioned for full marks: the comparison with the expected value, and the number of trials. Twelve rolls giving four sixes proves nothing, because small samples vary wildly; three hundred rolls giving ninety-five is a different matter entirely.

An answer saying only "yes, because it landed on six a lot" will not score.

Predicting with expected frequency

Once you have a probability — theoretical or experimental — the predicted number of occurrences in a set of trials is:

expected frequency = probability × number of trials.

If a spinner lands on red with relative frequency 0.42, then in 500 spins you would expect 0.42 × 500 = 210 reds.

This is an average expectation, not a promise. Getting 198 or 223 reds would be entirely normal, and questions often ask you to say precisely that.

Note that the prediction is only as good as the probability behind it, so an expected frequency built on 20 trials is far shakier than one built on 2000.

Relative frequency tables

Questions often give a table of outcomes with frequencies and ask you to complete a relative frequency row.

Find the total number of trials first by adding all the frequencies, then divide each frequency by that total.

The relative frequencies should sum to 1, which is a free check on the arithmetic. If they do not, either a division or the total is wrong.

Where results are given in stages — after 50 trials, after 100, after 200 — calculate the relative frequency at each stage using the cumulative totals, and the sequence of values usually shows the estimate settling.

Estimating a population size

A neat application uses relative frequency in reverse, to estimate how many items a large group contains — the capture-recapture idea.

A sample is taken from a population and marked, then returned. A second sample is taken later, and the proportion of marked items in it estimates the proportion marked in the whole population.

If 50 fish are tagged and released, and a later catch of 80 fish contains 10 tagged ones, then the relative frequency of tagged fish is 10 ÷ 80 = 0.125. Since 50 tagged fish make up that proportion of the whole, the population is roughly 50 ÷ 0.125 = 400.

The estimate relies on assumptions worth stating: that the tagged fish mixed evenly back into the population, that none were lost, and that the second sample was random. Questions often ask for one of these.

Reading a graph of relative frequency

Some questions plot relative frequency against the number of trials, and the shape carries the message.

The line typically swings widely at the left, where only a handful of trials have happened, then flattens towards a steady value as the trials accumulate.

That settling is the visual form of the rule that more trials give a better estimate, and the value the line levels off at is the best estimate of the true probability.

A line still swinging at the right-hand end means too few trials have been done to draw any conclusion, which is exactly what a question asking you to comment is looking for.

Comparing two experiments

When two people experiment with the same object, the one with more trials has the more reliable estimate, and combining the results gives a better one still.

To combine, add the frequencies and add the trials, then divide — never average the two relative frequencies, since that ignores the different sample sizes.

If one person gets 12 successes in 50 trials and another 45 in 200, the combined estimate is 57 ÷ 250 = 0.228, which is not the same as averaging 0.24 and 0.225.

Worked examples

Example 1: Calculating and predicting

A spinner is spun 200 times and lands on blue 46 times. Estimate the probability of blue, and predict how many blues there would be in 500 spins.

The relative frequency is 46 ÷ 200 = 0.23.

For the prediction, multiply by the number of trials: 0.23 × 500 = 115 blues.

It is an estimate in two senses — the probability itself came from an experiment, and the prediction is an average rather than a guarantee.

Example 2: Judging bias

A die is rolled 600 times and gives a four on 140 occasions. Is the die biased?

A fair die would give a theoretical probability of 1/6, so the expected number of fours in 600 rolls is 100.

The relative frequency here is 140 ÷ 600 = 0.233, compared with the fair value of about 0.167.

That is well above what would be expected, and it is based on 600 trials, which is a large enough sample for the estimate to be reliable.

So there is good evidence that the die is biased towards four.

Both elements are needed: the comparison against 1/6, and the point about the number of trials.

Example 3: A relative frequency table

A four-sided spinner is spun 250 times: red 60, blue 85, green 70, yellow 35. Complete the relative frequencies and comment on whether the spinner is fair.

Divide each frequency by 250:

Red 60 ÷ 250 = 0.24. Blue 85 ÷ 250 = 0.34. Green 70 ÷ 250 = 0.28. Yellow 35 ÷ 250 = 0.14.

Check: 0.24 + 0.34 + 0.28 + 0.14 = 1. ✓

A fair four-sided spinner would give each colour about 0.25. Blue is well above that and yellow well below, over 250 spins, so the spinner appears biased — it favours blue and rarely gives yellow.

Common mistakes and how to avoid them

Dividing by the wrong total. The denominator is the number of trials, not the number of other outcomes.

Calling relative frequency the exact probability. It is an estimate, and its quality depends on the number of trials.

Judging bias without mentioning the sample size. A small sample proves nothing, however extreme.

Saying "do more trials" without explaining why. More trials make the relative frequency settle closer to the true probability.

Averaging two relative frequencies. Combine by adding frequencies and adding trials, then dividing.

Treating an expected frequency as a guarantee. It is an average across many repetitions.

Using relative frequency for a fair object. If the outcomes are equally likely, calculate the probability theoretically instead.

Exam technique for "Relative Frequency"

Write the total number of trials down first, since it is the denominator for everything that follows and often carries a mark.

Check that a completed set of relative frequencies sums to 1 before moving on.

When judging fairness, always give three things: the theoretical value, the experimental value, and the number of trials.

Use the word estimate when describing a probability found from an experiment.

For predictions, show the multiplication and state that the answer is what you would expect on average.

If asked how to improve an estimate, say to increase the number of trials and explain that the relative frequency then settles closer to the true probability.

Quick revision summary

Relative frequency is what you use when you cannot reason the probability out — for a drawing pin or a bent coin, only evidence will do.

Relative frequency = frequency ÷ number of trials, always between 0 and 1, and always an estimate.

Use theoretical probability when outcomes are equally likely — the words fair or unbiased are the signal — and experimental probability otherwise.

More trials make the estimate better, because the relative frequency settles closer to the true probability. Say that, rather than just "do more trials".

To judge bias, compare the relative frequency with the theoretical value and state the number of trials. Both are needed.

Expected frequency = probability × number of trials, and it is an average rather than a guarantee.

In a table, relative frequencies should sum to 1. To combine two experiments, add the frequencies and add the trials — never average the two relative frequencies.