Matemáticas

What Is Expected Frequency?

A biased spinner spun 50 times, the 17.5 that is a correct answer, and the check that catches a slip: expected frequencies add back to the number of trials.

Expected frequency

Expected frequency is the number of times an outcome should occur on average, found by multiplying its probability by the number of trials.

También llamado: Expected number, Expected value of a frequency
Dónde lo encuentra el alumnado: Grade 7 probability, in questions phrased "how many times would you expect...", and again on GCSE and IGCSE papers alongside relative frequency.

La respuesta corta
Expected frequency is how many times an outcome should happen on average: its probability multiplied by the number of trials. A spinner with P(green) = 0.35 spun 50 times has an expected frequency of 17.5 greens. That is an average, so it need not be a whole number.

Un ejemplo

0.35 × 50 = 17.5

A biased spinner lands on red with probability 0.25, blue with 0.4 and green with 0.35. Spun 50 times, the expected frequencies are 12.5 red, 20 blue and 17.5 green. Nobody will ever record 17.5 greens. The figure is the average of the number of greens you would get if the 50 spins were repeated many times over, and as an average it is under no obligation to be whole.

Multiply, and do not round it away

The calculation is one multiplication, so the interesting part is what to do with the answer. Students often round 17.5 to 18 on the grounds that you cannot have half a spin, and that instinct throws information away.

An expected frequency is a long-run average, not a prediction of one experiment. Rounding is only right when the question asks for something that must be whole — the number of prizes to order, the number of chairs to put out — and then you should say why you rounded and which way. Where a question simply asks how many you would expect, 17.5 is the answer.

The check almost nobody does

The three expected frequencies from the spinner are 12.5, 20 and 17.5, and they add to 50 — the number of trials. That is not a coincidence and it is not luck. The probabilities add to 1, so multiplying each by 50 and adding gives 1 × 50.

It makes a free check on a whole table. If your expected frequencies do not sum to the number of trials, then either a probability is wrong, one has been missed, or a multiplication has slipped. On a question that supplies four or five outcomes, this catches errors faster than redoing the arithmetic.

Expected is not the same as will happen

You will very rarely get exactly the expected count, and that is not a failure of the calculation. Sixteen or nineteen greens in 50 spins is entirely unremarkable. Three greens would be a genuine reason to doubt the probability you were given.

It also helps to see where this sits next to relative frequency. Expected frequency travels from a known probability to a predicted count. Relative frequency travels the other way, from a recorded count to an estimated probability. The two are the same equation used in opposite directions, which is why a question that gives you an experiment and asks about 500 future trials wants both: estimate the probability from the results you have, then multiply by 500.

Preguntas frecuentes

Can an expected frequency be a decimal?

Yes. It is an average across many repetitions of the experiment, not a count from one of them, so there is nothing wrong with 17.5. The same idea lets a mean family size be 2.4 children. Only the actual result of a single experiment has to be a whole number.

Should I round my answer to a whole number?

Only when the context requires it. "How many times would you expect the spinner to land on green?" takes 17.5. "How many green prizes should the stall order?" takes a whole number, and you should round up and say so, since ordering 17 risks running short.

What is the difference between expected frequency and relative frequency?

They run in opposite directions. Expected frequency starts from a probability and predicts a count: probability × trials. Relative frequency starts from a count and estimates a probability: count ÷ trials. Questions often chain them, asking you to estimate a probability from 200 trials and then predict the result of 1,000.

How do I find expected frequency when I only have experimental results?

Use the relative frequency from your results as the probability, then multiply by the new number of trials. If 37 of 200 rolls gave a six, the estimated probability is 0.185, so in 1,000 rolls you would expect about 185. Say that it is an estimate based on the experiment.

If the expected frequency is 17.5, will I get 17 or 18?

Possibly neither. The expected frequency is the centre of a range of plausible results, not a forecast. Anything from about 13 to 22 would be ordinary for 50 spins at a probability of 0.35, and no single value in that range is being predicted over the others.

Última actualización 21 de agosto de 2026