Relative frequency and experimental probability GCSE both describe the same idea: estimating how likely an event is by repeating a trial many times and dividing the number of successful outcomes by the total number of trials. It is used when an event's true probability cannot be worked out just by reasoning, such as testing whether a die is biased.
What is relative frequency?
Relative frequency is the proportion of times an event actually happens during a set of repeated trials. It is calculated using the formula:
$$\text{relative frequency} = \frac{\text{number of successful trials}}{\text{total number of trials}}$$
Relative frequency is always a value between 0 and 1 (or expressed as a percentage between 0% and 100%), just like theoretical probability. The key difference is that relative frequency comes from real, observed data rather than from reasoning about equally likely outcomes.
What is experimental probability?
Experimental probability is the term used when relative frequency is treated as an estimate of the true probability of an event. In other words, once you calculate a relative frequency from a set of trials, you can use that value as your best estimate of the actual probability — this estimate is called the experimental probability.
The two terms describe the same calculation viewed from two angles: relative frequency is the raw result of an experiment, and experimental probability is what you call that result when you use it to estimate a true, underlying probability.
How is experimental probability different from theoretical probability?
Theoretical probability is calculated by reasoning about equally likely outcomes, without running any trials — for example, a fair coin has a theoretical probability of 0.5 for landing on heads, because there are two equally likely outcomes and one of them is heads. Experimental probability, by contrast, comes from actually carrying out trials and counting results.
| Type | How it is found | When it is used |
|---|---|---|
| Theoretical probability | Reasoning about equally likely outcomes, using the formula (favourable outcomes ÷ total possible outcomes) | When every outcome is known to be equally likely, such as a fair die or fair coin |
| Experimental probability | Repeating a trial and calculating relative frequency from the results | When outcomes are not known to be equally likely, or fairness needs to be tested |
The two values do not always match exactly, especially after a small number of trials — this difference is expected and does not necessarily mean anything is wrong with the experiment.
Worked example: testing whether a die is biased
A student suspects a six-sided die might be biased towards landing on 6. They roll the die 150 times and record 6 a total of 25 times of the recorded outcomes. Estimate the experimental probability of rolling a 6, and compare it to the theoretical probability for a fair die.
Step 1 — apply the relative frequency formula. $$\text{relative frequency of 6} = \frac{25}{150} = \frac{1}{6} \approx 0.167$$
Step 2 — find the theoretical probability for a fair die. $$P(6) = \frac{1}{6} \approx 0.167$$
Step 3 — compare the two values. The experimental probability (0.167) is extremely close to the theoretical probability for a fair die (0.167), so based on this data there is no strong evidence that the die is biased.
If instead the student had recorded 6 a total of 45 times out of 150 rolls, the relative frequency would be 45 ÷ 150 = 0.3, which is much higher than the theoretical 0.167 — that gap would be good evidence the die favours landing on 6.
How does the number of trials affect experimental probability?
Experimental probability becomes a more reliable estimate of the true probability as the number of trials increases. With only a small number of trials, relative frequency can swing a long way from the true probability just by chance — rolling a fair die only 6 times might easily give zero 6s or two 6s, neither of which reflects the true probability of 1/6 particularly well.
This idea is sometimes called the law of large numbers: as the number of trials gets larger, the relative frequency tends to settle down and get closer to the true, underlying probability. This is why GCSE questions often ask you to compare relative frequencies calculated after different numbers of trials, and to explain that the estimate based on more trials is generally more reliable.
Why does experimental probability matter beyond the exam?
Experimental probability underpins how insurers estimate the likelihood of a claim, how quality-control teams estimate the proportion of faulty items on a production line, and how scientists estimate outcomes that cannot be predicted by pure reasoning alone. Wherever a true probability cannot be calculated from first principles — because the situation is too complex, or fairness cannot be assumed — relative frequency from repeated trials gives the best available estimate.
Because relative frequency depends on how many trials were run, GCSE exam questions frequently test whether you understand that more trials give a more trustworthy estimate, and that a single small experiment should never be treated as conclusive proof of a die, coin, or spinner being biased or unbiased.
Frequently asked questions
Is relative frequency the same as experimental probability?
Yes, in GCSE maths the two terms are used interchangeably — relative frequency is the calculation (successful trials ÷ total trials), and experimental probability is the name given to that value when it is used as an estimate of the true probability. Some exam boards favour one term over the other, but both refer to the same underlying idea.
How many trials do you need for a reliable experimental probability?
There is no fixed number, but generally the more trials you carry out, the closer the relative frequency tends to get to the true probability, according to the law of large numbers. GCSE questions often compare results from a small number of trials against results from a much larger number to illustrate that more trials give a more reliable estimate.
What is the formula for relative frequency?
The relative frequency formula is: relative frequency = number of successful trials ÷ total number of trials. This gives a value between 0 and 1, which can also be written as a percentage or a fraction, and it is calculated in exactly the same way regardless of what the experiment involves.
Can experimental probability ever be more accurate than theoretical probability?
When a situation genuinely is not equally likely — such as a biased die, an unevenly weighted spinner, or a real-world event with unknown factors — experimental probability based on a large number of trials can give a more accurate picture than a theoretical probability that wrongly assumes fairness. Theoretical probability is only as good as the assumption of equally likely outcomes it is built on.
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