Empirical Samples

$P(\text{heads})=\frac12.$

This is a theoretical probability: it follows from the assumption that the coin is fair. The corresponding theoretical probability distribution assigns probability $\frac12$ to heads and $\frac12$ to tails.

We can also estimate probability by carrying out an experiment. If a coin produces 23 heads in 40 tosses, its relative frequency of heads is

$\frac{23}{40}=0.575.$

This is an empirical probability, because it comes from observed results. It need not equal the theoretical probability exactly.

Why repeated experiments give different results

Chance creates random variation. One set of 20 tosses might contain 13 heads, while another set of 20 tosses might contain only 8. Neither result is impossible for a fair coin, and the difference does not by itself show that the coin has changed.

Small samples are particularly variable because each result has a large effect. After 10 tosses, one additional head changes the relative frequency by much more than one additional head after 1,000 tosses.

This is also why an actual outcome, such as the profit from a probability game, may differ from its expected value. The expected value describes a long-run average, not a guaranteed result from a small number of plays.

What happens as sample size increases

As an unbiased experiment is repeated many times under the same conditions, its empirical probability generally becomes more stable and tends towards the underlying theoretical probability. For a fair coin, the relative frequency of heads tends towards $0.5$.

Data for Relative frequency settling towards theoretical probability

The movement is not necessarily smooth. The relative frequency can move away from the theoretical value temporarily, even after more trials. The important long-run pattern is that fluctuations usually become smaller relative to the total number of trials. A larger sample therefore generally gives a more reliable estimate, but it does not guarantee an exact match.

The same idea applies to a whole probability distribution. For a fair six-sided die, the empirical proportion for each face tends towards $\frac16$ as the number of rolls increases.

Why the sample must be unbiased

Increasing sample size helps only when the data are collected fairly. An unbiased experiment should not systematically favour particular outcomes. Conditions should be kept consistent, every result should be recorded, and the stopping point should not be chosen because a convenient result has just appeared.

For a population survey or inspection, the people or objects sampled should represent the population of interest. A very large but badly selected sample can still give a misleading estimate of a population characteristic.

For example, to estimate the proportion of faulty items in a delivery, selecting items randomly from throughout the delivery is more representative than testing only the easiest box to reach.

Comparing samples of different sizes

Compare relative frequencies, not just numbers of successes. Suppose one experiment records 34 heads and 8 tails, while another records 120 heads and 40 tails. Their estimates are

$\frac{34}{42}\approx0.810$

and

$\frac{120}{160}=0.75.$

The second estimate is based on 160 trials rather than 42, so it is normally the more reliable individual estimate. The first experiment having a larger relative frequency does not make it better evidence.

If several people test the same object under the same conditions, use all the raw results to obtain the strongest estimate. In the Edexcel example, the four experiments contain 300 heads and 100 tails altogether, so

$P(\text{heads})\approx\frac{300}{400}=0.75.$

Combining the counts correctly gives more weight to larger experiments. Simply averaging the four separate relative frequencies would give a small experiment the same influence as a large one.

Connecting empirical and theoretical probability

When a theoretical model is known, compare it with the experimental distribution. Large differences in a small sample may be explained by chance, whereas persistent differences in a large, unbiased sample may suggest that the model or its assumptions are unsuitable.

When the theoretical probability is unknown—perhaps because a coin is biased—the relative frequency from a large unbiased sample can be used as an estimate. The conclusion should remain appropriately cautious: it is an estimate supported by the available data, not certainty about what the next trial will produce.