When the true probability of an event is unknown, repeating an experiment many times and recording the results gives an estimate of that probability.

How do I calculate relative frequency?

A spinner is spun $200$ times and lands on red $70$ times:

Why is relative frequency only an estimate, and not the true probability?

Every trial involves an element of chance, so different runs of the same experiment will almost never give exactly the same relative frequency. As the number of trials increases, though, the relative frequency tends to settle down and get closer to the true, underlying probability — this is why examiners always look for a large number of trials before treating a relative frequency as trustworthy.

How do I decide whether an object like a coin or spinner is biased?

Compare the observed relative frequency with the value expected if every outcome were equally likely (fair). A noticeably higher or lower result suggests bias — but only if the number of trials is large enough to trust.

A fair spinner with equal sectors would land on red about $\dfrac{50}{200}=0.25$ of the time. Since the observed $0.35$ is clearly higher, the spinner appears to be biased towards red.

How do I use relative frequency to predict future outcomes?

Multiply the relative frequency (treated as an estimated probability) by the new number of trials:

Using $0.35$ from above, the expected number of reds in $500$ future spins is $0.35\times500=175$.

How is expected frequency different from what will actually happen?

Expected frequency is a prediction based on an average rate, not a guarantee — an actual run of $500$ spins could easily land on red $170$ or $180$ times rather than exactly $175$. Examiners accept "expected frequency" as a best estimate; it should never be presented as a certainty.

Worked Example

A biased six-sided die is rolled $150$ times, and a six is rolled $40$ times.

(a) Calculate the relative frequency of rolling a six.

(b) Estimate how many sixes would be rolled in $450$ rolls.

(c) Comment on whether the die appears biased, given that a fair die would be expected to show a six on $\dfrac{1}{6}$ of rolls.

1. (a) $\text{Relative Frequency} = \dfrac{40}{150} = 0.2\overline{6} \approx 0.27$
2. (b) $\text{Expected Frequency} = 0.267\times450 \approx 120$ sixes.
3. (c) A fair die would show a six roughly $\dfrac{1}{6}\approx0.167$ of the time. Since the observed relative frequency ($\approx0.27$) is considerably higher, the die does appear to be biased towards rolling a six.