Relative Frequency and Expected Frequency

When we can't list all outcomes and assign equal probabilities — for example with a biased coin or a complex real-world event — we estimate probability from experimental data. The result is called the relative frequency.

Relative frequency $= f/n$: divide the number of times the event occurred ($f$) by the total number of trials ($n$).

• The result is always between 0 and 1 — it IS a probability estimate.
• With few trials, relative frequency fluctuates widely. With many trials, it stabilises near the theoretical value.
• Relative frequency is also called experimental probability.

Experimental (relative) frequency = number of times event occurred ÷ total number of trials. It is an estimate of probability based on data. As trial count increases, experimental frequency approaches theoretical probability.

Pause — copy the relative frequency formula (experimental probability = number of occurrences ÷ total trials) and note it is an estimate of the true probability — the estimate becomes more accurate as the number of trials increases into your book.

Experimental (relative) frequency = occurrences ÷ total trials is an estimate of probability based on real data. This estimate is imprecise for small trial counts but the law of large numbers guarantees it converges toward the theoretical probability as the number of trials increases without limit — not in any finite trial, but in the long run.

One of the most important results in probability is that relative frequencies converge to theoretical probabilities as the number of trials increases. This is called the law of large numbers.

The idea: Run an experiment many times. Plot the relative frequency after each trial. It will jump around at first, then settle closer and closer to the theoretical probability $P(E)$.

• Implication: Small samples can mislead. A coin landing 8 heads in 10 flips does not mean the coin is biased — this has a reasonable probability of happening by chance.
• Practical cutoff: There's no exact number, but in most HSC contexts, relative frequency from at least a few hundred trials is considered a reasonable estimate of probability.

Law of large numbers: as the number of trials increases, the experimental (relative) frequency of an event gets closer and closer to the theoretical probability. It does not guarantee a result in any finite number of trials.

Pause — copy the law of large numbers: as the number of trials increases, the relative frequency of an event approaches its theoretical probability — and note it applies to the long run, not any specific finite sample: short-run results can still vary widely into your book.

The law of large numbers says relative frequency converges to theoretical probability as n grows. Working in the forward direction: if you know the probability and the number of trials, expected frequency = P × n gives the average number of times the event should occur. For example, P(6) = 1/6 on a die, so in 120 rolls the expected frequency of a 6 is (1/6) × 120 = 20.

If you know the theoretical probability of an event and you're about to run an experiment, you can calculate how many times you expect the event to occur: $E = n \times p$.

Formula: $E = n \times p$, where $n$ = number of trials, $p$ = probability per trial.

• Interpretation: $E$ is the average number of times you expect the event over many repetitions of the $n$ trials. It's not guaranteed for any single run.
• Non-integer $E$: $E$ can be a decimal. $E = 33.3$ means "on average, about 33 times per 100 trials."

Expected frequency = theoretical probability × number of trials. E.g., P(6) = 1/6 on a die; in 120 rolls, expected frequency of 6 = (1/6) × 120 = 20. Expected frequency is a prediction; actual frequency will vary.

Pause — copy the expected frequency formula: expected frequency = P × n, with one worked example (e.g., P(head) = 0.5, n = 80 flips → expected 40 heads), and note it is a long-run prediction — actual frequency in any experiment will vary into your book.