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Cambridge IGCSE Mathematics — 0580 Extended

Topic 8.2: Probability — Relative & Expected Frequencies

When you cannot list equally likely outcomes — a drawing pin, a biased spinner, a real population — you estimate probability from an experiment. That estimate is the relative frequency. Once you have a probability, you can predict how often the event should happen in \(n\) trials: the expected frequency.

Relative frequency

\[ \text{relative frequency} = \dfrac{\text{frequency of the outcome}}{\text{number of trials}} \]

It is an estimate of the true probability. A short experiment can sit a long way from the truth; a long one usually sits close.

Method

  1. Read the frequency of the outcome and the total number of trials.
  2. Write frequency \(\div\) trials. Cancel, or convert to a decimal if asked.
  3. Treat the result as \(\mathrm{P}(\text{event})\) only as an estimate.

A spinner is spun 50 times. Red appears 18 times. Estimate \(\mathrm{P}(\text{red})\).

Four-colour spinner and a table of 50 spins showing 18 reds, so relative frequency of red is 18/50 = 0.36
Relative frequency of red \(= 18/50 = 0.36\). That estimates \(\mathrm{P}(\text{red})\), it does not prove the spinner is biased.

More trials, better estimate

As \(n\) grows, the relative frequency typically settles toward the true probability. One run of 10 spins that lands far from \(0.25\) is not evidence of bias on its own.

Graph of relative frequency against number of trials settling toward the true probability 0.25
The dashed line is the true probability. The experimental path wanders, then closes in.

Fair, biased and random

A fair spinner (or die, or coin) has equal chance on every equal sector. Biased means not fair — one outcome is favoured. Random means each trial is independent of the last: the spinner has no memory.

Fair four-colour spinner beside a biased spinner with a larger red sector
Look at the object itself to decide fair or biased. A short table of results is not a proof.

Expected frequency

\[ \text{expected frequency} = n \times \mathrm{P}(\text{event}) \]

It is a prediction, not a guarantee. The real count can sit either side of the expected value.

In a town, \(\mathrm{P}(\text{left-handed}) = 0.12\). How many left-handed people would you expect in a random sample of 200?

200 dots with 24 highlighted, showing expected frequency 200 times 0.12 equals 24
\(200 \times 0.12 = 24\). You would expect 24, not “exactly 24 every time”.

A spinner lands on blue 27 times in 90 spins. Estimate \(\mathrm{P}(\text{blue})\), then the expected number of blues in 300 spins.

Spinner experiment 27 blues in 90 spins giving estimate 0.3 and expected 90 blues in 300 spins
Estimate \(\mathrm{P}(\text{blue}) \approx 27/90 = 0.3\), then \(300 \times 0.3 = 90\).

Paper 2 (non-calculator)

\(80 \times 3/5 = 48\). Cancel first: \(80 \times 3 / 5 = 16 \times 3 = 48\). Do not expand into a decimal if the fraction is clean.

Try this

A fair cubical die is rolled 120 times. How many 6s would you expect?

Show answer
Answer

  1. Fair die, so \(\mathrm{P}(6) = 1/6\)

    \[ 120 \times \dfrac{1}{6} = 20 \]

Exam Traps

  • Relative frequency is an estimate of probability. It is not the theoretical probability, and a small sample that disagrees with \(1/4\) does not prove the spinner is biased.

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