Relative frequency and expected frequency
Estimating probability from data, and predicting outcomes.
Review these first
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 8.2.1Use relative frequency to estimate probability and calculate expected frequencies.
4 minute read
Relative frequency and expected frequency
Relative frequency
When a probability cannot be worked out theoretically — an unfair dice, a drawing pin landing point up — it is estimated from experiment: relative frequency = number of times the event happened / total number of trials
Reliability
The estimate improves as the number of trials increases. With 10 trials it may be far from the true value; with 1000 it will usually be close. If asked how to improve an estimate, the answer is always to increase the number of trials.
Theoretical versus experimental
- Theoretical probability comes from the structure of the situation — a fair dice gives 1/6 for each face.
- Experimental (relative frequency) comes from what actually happened.
If the two differ substantially over many trials, that is evidence the object is biased. Small differences over few trials mean nothing.
Expected frequency
To predict how many times an event will occur: expected frequency = probability × number of trials If P(red) = 0.3 and there are 200 spins, the expected number of reds is 0.3 × 200 = 60. Expected frequency is a prediction, not a guarantee — getting 58 or 63 reds would be entirely unremarkable. Questions often reserve a mark for saying so.
Think of it like this
Relative frequency is an opinion poll on a dice: ask it ten times and the result is noisy, ask it a thousand times and you can trust the answer. More trials means a smaller margin of error.
Worked examples
Method, step by step
A spinner is spun 200 times and lands on blue 46 times. Estimate the probability of blue, and predict how many blues in 500 spins.
- 1Relative frequency = 46/200 = 0.23.
- 2Expected frequency = probability × number of trials.
- 3= 0.23 × 500 = 115.
P(blue) ≈ 0.23; about 115 blues expected in 500 spins
Common misconceptions
- Treating expected frequency as exactly what will happen. It is the most likely long-run average, not a promise.
- Concluding a dice is biased from a handful of throws. Small samples vary a great deal by chance.
- Confusing relative frequency with frequency. Relative frequency is the proportion, so it lies between 0 and 1.
In the exam
- If asked how to make an estimate more reliable, the expected answer is "carry out more trials".
- When comparing experimental with theoretical probability, comment on the **number of trials** before declaring bias.