Estimating probability using relative frequency, the effect of more trials, comparing experimental and theoretical probability, and finding expected outcomes.

What this dot point is asking

AQA wants you to estimate probability from experimental data using relative frequency, understand why more trials give a better estimate, compare experimental and theoretical probability to judge bias, and calculate expected outcomes. This is the practical side of probability: when outcomes are not equally likely (a drawing pin, a biased spinner), you must measure rather than count.

Relative frequency as an estimate

When outcomes are not equally likely, you cannot use counting; instead you run trials and record results.

If a drawing pin lands point-up $63$ times out of $150$ drops, the relative frequency is $\dfrac{63}{150} = 0.42$, so the estimated probability of landing point-up is $0.42$. There is no way to find this from symmetry; only experiment gives it.

The effect of more trials

A relative frequency from a small number of trials can be unreliable; a few unusual results swing it a lot. As the number of trials grows, the relative frequency stabilises and converges toward the true probability. This is the law of large numbers. In an exam, if asked which of two estimates is more reliable, choose the one based on more trials, and recommend more trials to improve any estimate.

Comparing experimental and theoretical probability

For a fair object you can also compute the theoretical probability (for a fair dice, $P(6) = \tfrac{1}{6} \approx 0.17$). Comparing this with the experimental relative frequency tests for bias: if the experimental value is consistently well above or below the theoretical value across many trials, the object is probably biased. A small difference over few trials, though, is likely just chance. Always weigh the size of the difference against the number of trials.

Expected outcomes

Expected frequency is the probability multiplied by the number of trials. If the probability of a faulty component is $0.03$, then in a batch of $2000$ you expect about $0.03 \times 2000 = 60$ faulty ones. This is an estimate of the long-run average, not a precise count, and it underlies quality-control and risk problems.

Combining results from several experiments

When data comes in batches, the best estimate uses all of it combined, not the average of the separate estimates. If one set of $40$ spins gives $14$ reds and another set of $60$ spins gives $25$ reds, the pooled estimate is $\dfrac{14 + 25}{40 + 60} = \dfrac{39}{100} = 0.39$, using the total reds over the total spins. This is more reliable than averaging $0.35$ and $0.42$, because it correctly weights the larger experiment. Examiners use this to test whether you understand that more data means a better estimate.

Relative frequency versus theoretical probability

It helps to keep the two ideas distinct. Theoretical probability comes from the structure of a fair object (a fair coin gives $\tfrac{1}{2}$), known before any trial. Relative frequency comes from observed results and is the only option when an object may be biased or its structure is unknown, such as a drawing pin or a real-world event like a bus arriving late. In practice, relative frequency from a large experiment is how you would estimate the probability of something that has no obvious symmetry, and the law of large numbers is the bridge that says the two agree for a fair object given enough trials.