What you'll learn
Relative frequency (experimental probability) uses results from trials to estimate probabilities. In this guide you will learn how to calculate relative frequency, how it estimates probability, how the estimate improves with more trials, how to use it to predict outcomes, and how to test whether something is fair. This complements theoretical probability at GCSE.
Key terms and definitions
Relative frequency — the number of times an event happened divided by the total number of trials.
Experimental probability — probability estimated from experimental results (the same as relative frequency).
Theoretical probability — probability worked out by reasoning about equally likely outcomes.
Trial — one repetition of an experiment.
Expected frequency — the predicted number of times an event will occur, found by multiplying probability by the number of trials.
Core concepts
Calculating relative frequency
Relative frequency is calculated as:
relative frequency = frequency of the event ÷ total number of trials
For example, if a drawing pin lands "point up" 36 times out of 100 throws, the relative frequency is 36 ÷ 100 = 0.36. It is always a value between 0 and 1.
Relative frequency as an estimate of probability
When outcomes are not equally likely (like a biased coin or a drawing pin), we cannot use theoretical probability, so we estimate the probability from experiments. The relative frequency gives this experimental probability.
More trials give a better estimate
The more trials you carry out, the closer the relative frequency tends to get to the true probability. A small number of trials can be misleading; large numbers give a more reliable estimate. This is why experiments are repeated many times.
Predicting outcomes (expected frequency)
Once you have a probability, you can predict how many times an event will happen in future trials:
expected frequency = probability × number of trials
For example, if the probability is 0.36 and you throw the pin 250 times, you expect about 0.36 × 250 = 90 "point up" results.
Testing for fairness
Comparing the relative frequency with the theoretical probability can suggest whether something is fair or biased. If a die's relative frequency for a six is close to 1/6 over many throws, it is probably fair; if it is much higher or lower over many trials, it may be biased.
Worked examples
Example 1: Relative frequency
A spinner lands on red 18 times in 60 spins. What is the relative frequency of red?
18 ÷ 60 = 0.3.
Example 2: Expected frequency
The probability of a faulty bulb is 0.05. How many faulty bulbs are expected in 400?
0.05 × 400 = 20 faulty bulbs.
Example 3: Judging fairness
A coin shows heads 540 times in 1000 flips. Is it likely fair?
Relative frequency = 0.54, close to the theoretical 0.5 over many trials, so it is probably fair (slight variation is expected).
Common mistakes and how to avoid them
Dividing the wrong way. Relative frequency = event frequency ÷ total trials.
Trusting too few trials. Small samples can be misleading; more trials give better estimates.
Confusing experimental and theoretical probability. Use relative frequency when outcomes aren't equally likely.
Forgetting to multiply for predictions. Expected frequency = probability × number of trials.
Expecting exact matches. Real results vary around the true probability.
Exam technique for Relative Frequency
Use the formula — event frequency ÷ total trials.
State that more trials improve the estimate.
Predict with expected frequency = probability × trials.
Compare with theoretical probability to judge fairness.
Allow for natural variation rather than expecting exact values.
Quick revision summary
Relative frequency (experimental probability) estimates probability from results: relative frequency = frequency of the event ÷ total number of trials, always between 0 and 1 (a pin landing point-up 36 times in 100 → 0.36). It is used when outcomes are not equally likely, so theoretical probability cannot be calculated. The more trials you do, the closer the relative frequency gets to the true probability, which is why experiments are repeated many times. To predict future outcomes, use expected frequency = probability × number of trials (0.36 × 250 ≈ 90). Comparing relative frequency with the theoretical probability over many trials helps judge whether something is fair or biased. The common errors are dividing the wrong way, trusting too few trials, confusing experimental with theoretical probability, forgetting to multiply for predictions, and expecting exact matches. Use the formula, do many trials, predict with expected frequency, compare against theory for fairness, and remember real results vary around the true value.