Mathématiques

What Is Relative Frequency?

Worked from 200 recorded rolls and compared with the theoretical 1/6, plus why you combine two students' results by adding totals, not averaging fractions.

Relative frequency

Relative frequency is the number of times an outcome occurred divided by the total number of trials, used as an estimate of that outcome's probability.

Aussi appelé: Experimental probability, Estimated probability
Où les élèves le rencontrent: Grade 7 probability, the first time a class tests whether a die or spinner is fair, and again in GCSE and IGCSE questions about estimating probability from a table.

La réponse en bref
Relative frequency is the number of times an outcome happened divided by the number of trials, used to estimate a probability you cannot calculate. Thirty-seven sixes in 200 rolls gives 37/200 = 0.185. More trials steady the estimate, which is why the largest experiment gives the best one.

Un exemple

37 sixes in 200 rolls → 37 ÷ 200 = 0.185

A fair die would give a probability of 1/6, about 0.167, so 200 rolls would be expected to produce about 33 sixes. Thirty-seven is more than that, and it is nothing to write home about: the count varies from experiment to experiment, and a gap of four in 200 rolls is ordinary. The relative frequency estimates the probability; it does not measure it.

One is a measurement, the other a calculation

Theoretical probability is worked out from the structure of the thing — six faces, so 1/6. Relative frequency is worked out from what actually happened when you tried it. They answer the same question by different routes, and only one of them can be used on an object you cannot see inside.

That is when relative frequency stops being a classroom exercise. Nobody can calculate from first principles the probability that a drawing pin lands point up, or that a particular bus is late. You collect trials and count. For a fair die, comparing the two is a way of checking the experiment; for a drawing pin, the experiment is the only source there is.

Why more trials narrow the gap

In 20 rolls, one extra six moves the estimate by 0.05. In 200 rolls it moves it by 0.005, and in 2,000 by 0.0005. The estimate steadies because the divisor grows, so each individual trial has less pull on the answer.

Notice what this does and does not promise. The gap between the number of sixes and the expected number can grow as you roll more — being 15 sixes away from expectation after 2,000 rolls is unremarkable. It is the proportion that tightens, not the count. This is why the answer to "how many trials are enough" is never a fixed number, and why every extra trial helps a little.

Combining two sets of results

A standard exam question gives two students' experiments and asks for the best estimate. Student A records 12 successes in 50 trials, or 0.24. Student B records 38 in 150 trials, about 0.253.

The correct method is to pool the raw counts: 50 successes in 200 trials, giving 0.25. Averaging the two fractions gives 0.247, which is wrong because it treats Student A's 50 trials as carrying the same weight as Student B's 150. The same principle answers the other version of the question — the single best estimate is the one from the largest number of trials, because the number of trials is what makes an estimate trustworthy.

Questions fréquentes

What is the difference between relative frequency and probability?

Probability is what should happen in the long run; relative frequency is what did happen in the trials you ran. Relative frequency is used as an estimate of probability, and it gets closer to it as the number of trials increases, but the two are rarely equal in any one experiment.

Does a die that gives 37 sixes in 200 rolls have a bias?

There is no real evidence of it. A fair die would produce about 33 sixes on average across 200 rolls, and results in the low thirties or high thirties happen routinely by chance. Something like 60 sixes in 200 rolls would be a reason to look at the die more closely.

How many trials are enough?

There is no fixed number, and any answer claiming one is inventing it. More trials always give a steadier estimate, and how many you need depends on how precise the answer has to be. Class experiments usually pool everyone's results for exactly this reason.

Do relative frequencies add up to 1?

Yes, across all the possible outcomes. Every trial produced exactly one outcome, so the counts add to the number of trials and the fractions add to 1. It is a quick check on a completed table — if they do not, an outcome has been miscounted or one has been left out.

Can relative frequency prove a spinner is fair?

No. It can build evidence against fairness, and it can fail to find any, but it cannot prove fairness — a slight bias will always be within the range of ordinary variation for some number of trials. The honest phrasing is that the results are or are not consistent with a fair spinner.

Dernière mise à jour 21 août 2026