Experimental Probability and Relative Frequency
Master Experimental Probability and Relative Frequency for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. Relative frequency estimates probability from the results of trials and improves as the number of trials increases.
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Topic overview
Experimental probability is estimated from the results of trials rather than calculated from theory. It is also called relative frequency.
The formula is \(\text{relative frequency} = \frac{\text{number of times the event occurred}}{\text{total number of trials}}\).
The more trials carried out, the closer the experimental probability tends to get to the true probability. This is why a small number of trials gives an unreliable estimate. Comparing experimental with theoretical probability is how bias is detected: a large difference over many trials suggests the object is biased, though a difference over few trials proves nothing.
Revision notes
The formula
\(\text{Relative frequency} = \frac{\text{number of successes}}{\text{total number of trials}}\).
This estimates the probability from what actually happened rather than from theory. It is the only method available when outcomes are not equally likely.
Why more trials are better
As the number of trials increases, the relative frequency tends towards the true probability.
A small number of trials gives an unreliable estimate, because chance variation has a large effect. This is why questions asking how to improve an estimate expect 'carry out more trials'.
Detecting bias
Compare the experimental probability with the theoretical probability.
A large difference over many trials suggests the object is biased. A difference over few trials proves nothing, because it could easily arise by chance.
Key points
- Experimental probability comes from trials.
- It is also called relative frequency.
- \(\frac{\text{successes}}{\text{total trials}}\).
- More trials give a more reliable estimate.
- It is the only method for biased objects.
- Comparing with theory detects bias.
Worked examples
Example 1
A dice is rolled 200 times and shows a six 45 times. Work out the relative frequency of a six. [2 marks]
Working
Example 2
Explain how to improve the reliability of an experimental probability. [2 marks]
Working
Example 3
A fair dice should show a six with probability \(\frac{1}{6}\). The relative frequency over 600 rolls is 0.25. Comment on this. [2 marks]
Working
Common mistakes
Confusing experimental with theoretical probability.
Experimental comes from trials; theoretical from counting outcomes.
Concluding bias from few trials.
A difference over few trials could easily arise by chance.
Dividing by the number of successes.
Divide by the total number of trials.
Saying more trials guarantee the true probability.
They make the estimate more reliable, not exact.
Exam tips
- Divide by the total number of trials.
- Say 'carry out more trials' to improve reliability.
- Compare with theoretical probability to detect bias.
- Note how many trials before concluding bias.
Key terms
- Experimental probability
- Probability estimated from trials.
- Relative frequency
- Another term for experimental probability.
- Trial
- One repetition of an experiment.
- Bias
- A tendency for some outcomes to be more likely.
Related topics
Written and reviewed against the current AQA and Edexcel specifications. Spotted an error? Let us know.