Estimating probability from experimental data using relative frequency, comparing experimental and theoretical probability, and calculating the expected number of outcomes from a probability.

What this dot point is asking

Edexcel expects you to estimate probability from experimental data using relative frequency, to compare experimental results with theoretical probability to judge fairness, and to calculate the expected number of times an outcome should occur. This is where probability meets data: theory predicts, experiments test, and the two are compared.

Relative frequency

When outcomes are not equally likely, or when we cannot calculate a theoretical probability, we estimate it from an experiment.

So if a drawing pin lands "point up" $72$ times in $120$ drops, the relative frequency is $\dfrac{72}{120} = 0.6$, our best estimate of the probability. A single small experiment gives a rough estimate; thousands of trials give a much better one, because relative frequency settles towards the true probability over many trials.

Experimental versus theoretical probability

Theoretical probability comes from the structure of the situation (a fair die gives $\tfrac{1}{6}$ for each face). Experimental probability comes from data. Comparing them is how we test fairness.

If a coin is flipped $1000$ times and lands heads $720$ times, the relative frequency $0.72$ is far from the theoretical $0.5$, suggesting the coin is biased. Small differences are expected from chance, so the judgement should be cautious: a relative frequency close to the theoretical value supports fairness, while a large gap over many trials suggests bias.

Expected number of outcomes

The expected number predicts how many times an outcome should occur, given its probability.

The expected value is also used the other way round: given an expected count and the number of trials, you can find the probability by dividing.

Why more trials matter

A key idea Edexcel tests is that relative frequency becomes a better estimate as the number of trials grows. With only ten spins of a spinner, a relative frequency of $0.3$ could easily be out by a lot; with ten thousand spins, a relative frequency of $0.3$ is strong evidence that the true probability is close to $0.3$. This is the "law of large numbers" in plain terms, and it is why an experiment to test fairness should use many trials. A question may give two experiments of different sizes and ask which estimate to trust; the larger experiment is the more reliable.

Try this

Q1. A coin is flipped $80$ times and lands heads $36$ times. Work out the relative frequency of heads. [2 marks]

• Cue. $\dfrac{36}{80} = 0.45$.

Q2. The probability a train is late is $0.1$. Over $250$ journeys, how many are expected to be late? [2 marks]

• Cue. $0.1 \times 250 = 25$ journeys.