Fair
A fair six-sided die gives each face probability 16. A fair coin gives each side probability 12.
CAMBRIDGE O LEVEL · 2025–2027
Relative and expected frequencies.
Use experimental evidence to estimate chance and probability to predict a frequency.
BY THE END
Calculate relative frequency · interpret fair, biased and random · estimate expected frequencies
01
Estimate probability from what happened
A theoretical probability uses a model such as a fair die. Sometimes no reliable model is known, so we observe repeated trials and use the results to estimate the chance.
Relative frequency is the share of trials in which an event occurred. For example, 18 red results in 50 trials gives estimate 1850. Repeating an experiment can change that estimate; it is evidence about probability, not a guarantee of the next result.
When the true probability is unknown, use experimental results. Relative frequency is the proportion of trials in which the event occurred.
Relative frequency = frequency of the eventnumber of trials
WORKED EXAMPLE
A spinner lands on red 38 times in 100 spins. Estimate P(red).
Estimated P(red) = 38100 = 0.38
This is an estimate from the experiment, not proof that the true probability is exactly 0.38.
SPINNER EXPERIMENT · FOUR EQUAL SECTORS
100 spins recorded. Initial data: 100 recorded spins; new spins are simulated with equal chances.
Estimated P(A) = 24100 ≈ 0.240
Pause and explainDoes an estimated probability of 0.4 guarantee four successes in the next ten trials?
No. It suggests a long-run proportion, while a small future group can vary by chance.
02
Fair, biased and random mean different things
A random result is uncertain, but it need not give every outcome equal chance. Drawing from a bag with mostly red counters is random even though red is much more likely than blue.
Fairness describes equal chances in the intended model, while bias describes favouring some outcomes compared with that fair model. A short run of uneven results can arise by chance. More evidence helps distinguish ordinary variation from a persistent imbalance.
Biased
A biased device favours some outcomes compared with the fair model. Unequal results in a short experiment alone do not prove bias.
A larger, representative sample usually supports a better estimate. A large sample selected in a biased way can still give a misleading result for a whole population.
Pause and explainCan a fair coin produce five heads in a row?
Yes. A short uneven run is possible even though each toss has equal chances.
03
Predict a frequency using a probability
Expected frequency turns a probability into an anticipated count over many trials or a population. If each item has chance 0.2 of a property, that property is expected in about one fifth of a large collection.
Multiply the total number by the probability or estimated proportion. This gives an expectation, not a promise about one particular sample. If the probability comes from a survey, the survey must reasonably represent the population for the prediction to be useful.
Expected frequency = number of trials × probability
WORKED EXAMPLE
A machine produces a defective item with probability 0.03. Estimate the number of defective items in 800 items.
800 × 0.03 = 24
Expect about 24 defective items. This does not guarantee exactly 24 in this batch.
WORKED EXAMPLE
In a representative sample, 45 of 150 people cycle to work. Estimate how many of a population of 2400 cycle to work.
Estimated P(cycles) = 45150 = 0.3
Expected number = 2400 × 0.3 = 720
Expected number = 2400 × 0.3 = 720
The estimate assumes the sample reflects the population.
Pause and explainWhat does an expected frequency of 12 mean?
It is the predicted or average count for the stated group size, not a certainty that exactly 12 will occur.
04
Your turn. Estimate, then interpret.
Distinguish estimating a probability from predicting a count. Divide observed event frequency by trials for the first; multiply a probability by a new total for the second.
Describe the answer as an estimate or expectation where appropriate. Mention sample quality when applying experimental results to a wider population.
1
A coin lands heads 92 times in 200 tosses. Estimate P(heads).
Reveal worked answer
92200 = 2350 = 0.46.
2
Using P(heads) = 0.46, estimate the number of heads in 500 further tosses.
Reveal worked answer
500 × 0.46 = 230 heads. The actual number may differ.
3
A fair die is rolled 90 times. How many sixes are expected?
Reveal worked answer
90 × 16 = 15.
4
A representative sample has 18 left-handed people out of 120. Estimate the number in a population of 2000.
Reveal worked answer
18120 = 0.15; 2000 × 0.15 = 300.
5
A fair coin gives 7 heads in 10 tosses. Must it be biased?
Reveal worked answer
No. Random variation can give unequal counts. More trials provide stronger evidence about its probabilities.