Pearson Edexcel · GCSE · Mathematics
Course version: 1MA1 · Version from Mar 2026
Understand that empirical unbiased samples tend towards theoretical…
Full specification wording
Probability: understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size
The more trials you run, the closer the relative frequency from an unbiased experiment settles towards the theoretical probability.
Checked against the Pearson Edexcel 1MA1 specification (opens in a new tab) on .
- Point 5 of 9
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
The more trials you run, the closer the relative frequency from an unbiased experiment settles towards the theoretical probability. That is why a result from 500 throws is a better estimate than one from 20 throws, and why two short experiments can disagree with each other, and with the theory, without either being wrong. Questions give a table of results after different numbers of trials and ask which estimate to trust, or ask you to pool the results by adding all the successes and dividing by all the trials. Ten throws of a coin giving 7 heads is a relative frequency of 0.7, which is far too short a run to show bias, whereas a thousand throws giving 700 heads would be strong evidence of it. The relative frequency settles closer and closer to the theoretical value rather than locking onto it.
What earns the marks
- Choosing the estimate that comes from the largest number of trials, with a reason given.
- Pooling results as total successes over total trials, rather than averaging the separate fractions.
- A relative frequency written correctly as a fraction, decimal or percentage.
- A comment linking closer agreement with the theoretical value to a larger sample.
Common mistakes and how to put them right
Averaging the relative frequencies of each stage instead of using the combined totals.
Claiming a small sample proves bias because its result is far from the expected one.
Expecting the running relative frequency to land exactly on the theoretical probability and stay there.
Choosing the estimate closest to the theoretical value rather than the one from the most trials.
Exam tips
- When asked which estimate is best, quote the number of trials in your reason. Saying it is bigger, with no number, is not a reason.
- If a table adds trials in stages, keep a running total column as you go, so the pooled figures are already written down when a later part needs them.
Try the lesson
The start of the Probability: understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size lesson, made from checked teaching notes for this specification point.
Have a go first
A fair coin is tossed 10 times and lands on heads 7 times. The coin is then tossed 1000 times. Which statement best describes what is likely to happen to the proportion of heads as the number of tosses increases?
Listen
The more trials you run, the closer the relative frequency from an unbiased experiment settles towards the theoretical probability. That is why a result from 500 throws is a better estimate than one from 20 throws, and why two short experiments can disagree with each other, and with the theory, without either being wrong. Questions give a table of results after different numbers of trials and ask which estimate to trust, or ask you to pool the results by adding all the successes and dividing by all the trials. Ten throws of a coin giving 7 heads is a relative frequency of 0.7, which is far too short a run to show bias, whereas a thousand throws giving 700 heads would be strong evidence of it. The relative frequency settles closer and closer to the theoretical value rather than locking onto it. A strong answer earns marks for these points. choosing the estimate that comes from the largest number of trials, with a reason given. pooling results as total successes over total trials, rather than averaging the separate fractions. a relative frequency written correctly as a fraction, decimal or percentage. a comment linking closer agreement with the theoretical value to a larger sample. Exam tip: When asked which estimate is best, quote the number of trials in your reason. Saying it is bigger, with no number, is not a reason.
Key words
Say what each word means before you turn the card.
sample size
Try answering aloud or in your head before turning the card.
relative frequency estimates
Try answering aloud or in your head before turning the card.
unbiased samples
Try answering aloud or in your head before turning the card.
pooling results
Try answering aloud or in your head before turning the card.
See every definition
- sample size
- The number of trials run; a larger one makes the relative frequency settle closer to the theoretical probability.
- relative frequency estimates
- A value from an experiment, written as a fraction, decimal or percentage, that settles closer to the theoretical value as trials increase.
- unbiased samples
- Results from an experiment with no bias, whose relative frequency tends towards the theoretical probability distribution as the sample size increases.
- pooling results
- Combining stages by taking total successes over total trials, rather than averaging the separate fractions.
There is more to this lesson
The full lesson goes on to check your understanding and help you earn the marks. Create a free account to carry on, with your progress saved.
A week of Premium from your first lesson · Then the free plan continues · No card, no charge
Test your understanding of Probability: understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size
Active recall beats re-reading. Take an instant quiz now to test what you’ve learned and lock it into long-term memory.
A week of Premium from your first lesson · Then the free plan continues · No card, no charge