Observed relative frequency
An experimental estimate uses completed trials from a specified process
- Design and record Define one complete trial and its success event. Keep conditions consistent and record every result
- Combine categories Add disjoint favourable frequencies. Include both endpoints when a question says inclusive
- Combine runs For the same experiment, add successes and trial counts separately; do not average unequal-sized run proportions
- Predict carefully Multiply an estimate by a future trial count for a predicted count. This is not a guarantee
- Assess uncertainty Zero or all successes in a finite sample do not prove impossibility or certainty. Larger samples do not repair biased methods
Worked example
The same cap-toss method gives 7 successes in 20 trials, then 11 successes in a further 30 trials. Find the pooled estimate and predict successes in 50 further trials
- \(\displaystyle s=7+11=18\)Add successful trials
- \(\displaystyle T=20+30=50\)Add every completed trial
- \(\displaystyle \widehat p=\frac{18}{50}=\frac{9}{25}\)Pooling gives equal weight to each trial
- \(\displaystyle 50\times\frac{9}{25}=18\)Predict 18 successes under the same conditions. The actual count can differ
Common mistakes
Difficulty levels
Questions about this topic
Can I estimate a probability before any trials?+
No. With zero trials there is no relative-frequency estimate
Must my estimate match someone else’s?+
No. Finite runs vary. Compare the trial definition and conditions as well as the counts
What if the true probability is unknown?+
Use a carefully designed experiment and its recorded relative frequency. Report the method, trial count and uncertainty rather than calling it exact
Can I pool every data set?+
Only when combining them answers the intended question. Different conditions or populations can change what the combined proportion means
Is every larger sample better?+
More representative, consistently collected data usually reduce random variation. A larger biased sample can still mislead
Is using the recorder proof I did the experiment?+
No. The tool calculates your entered counts; you must conduct and document the trials. A teacher or peer can review your method and record
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 9: Probability and statistics