Understand the idea.
Choose the rules before starting
Before starting, define one valid trial, the outcomes you will record and how many valid trials you will complete. For a coin toss, record the upper face. Agree in advance whether an edge landing or a toss outside the tray must be repeated.
Record every valid outcome immediately in a tally or frequency table, including unexpected results. Provide a row for each possible outcome. Finish at the planned trial count, so the result does not depend on stopping when your favourite pattern appears.
Keep the model unchanged and report honestly
Keep the chance procedure consistent. For bag draws using the original colour probabilities each time, record the colour, replace the counter, remix and choose without looking. Removing counters changes the available objects and can change the probability for the next draw.
Check that the outcome frequencies add to the valid trial total. Compare observed counts with the model's expected counts and report any differences honestly. A short unbalanced record may justify further investigation, but alone it does not establish that a device is unfair.
| Result | Frequency |
|---|---|
| Heads | 12 |
| Tails | 8 |
See the method in action.
Check an experimental record
Use the coin table. Find the number of trials and how far the heads count is above the fair-coin expectation.
| Result | Frequency |
|---|---|
| Heads | 12 |
| Tails | 8 |
Worked solution
- 1
Add both recorded outcome counts.
Each valid coin toss produces either heads or tails, so both rows contribute to the trial total.
- 2
Calculate the fair-coin expected heads count.
Heads has probability one half on each fair-coin toss.
- 3
Compare observed heads with expected heads.
The observed count is two above expectation; this difference does not by itself prove the coin unfair.
20 trials; heads are 2 above expectation.
Improve a repeated bag experiment
A student removes counters after each draw but uses the original colour probability every time. How should they change the procedure to keep that probability unchanged?
Worked solution
- 1
Identify why the original procedure changes the model.
Keeping a drawn counter out changes the bag's contents, so the original colour probability cannot generally be reused for every draw.
- 2
Record the colour, then replace the counter.
Replacement restores the original total and the original number of counters of each colour.
- 3
Remix and choose without looking before the next draw.
Mixing prevents the replaced counter's position from guiding selection, while blind selection avoids deliberately favouring a colour.
Replace and remix the counter after every draw.
Try it yourself.
Check your understanding
A 30-trial table contains frequencies 8, 11 and 10. What needs checking?
Show answer & reasoning
One trial is missing or a count is incorrect.
The frequencies total , which is below the stated trial count. Check the original tally for a missing result or incorrect count; do not invent an outcome to make the total fit.
Keep practising
6 worksheet questions · 16 marks
Read the questions online
- Describe one trial when testing the colour results of a spinner.2 marks
- An experiment records 9 red, 7 blue and 8 green results. How many valid trials are recorded?2 marks
- Why should a counter be returned and the bag remixed when each draw is meant to have the original probabilities?3 marks
- Use the coin table. State the observed and fair-model expected tails counts.2 marks
Example record of 20 coin tosses Result Frequency Heads 12 Tails 8 - A student only records heads and ignores tails. Explain why the record is incomplete.3 marks
- Plan 40 draws from a bag of 2 red and 3 blue equal-size counters. Include selection, replacement, recording and expected reds.4 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Describe one trial when testing the colour results of a spinner.2 marks
State both the action and the recorded result.
Choose a boundary rule in advance.
Spin once and record the colour where the pointer stops. - An experiment records 9 red, 7 blue and 8 green results. How many valid trials are recorded?2 marks
Add every frequency: .
24. - Why should a counter be returned and the bag remixed when each draw is meant to have the original probabilities?3 marks
Replacement restores the numbers of each colour.
Mix and choose without looking to avoid deliberate selection.
Returning restores the contents; remixing helps keep selection random. - Use the coin table. State the observed and fair-model expected tails counts.2 marks
Example record of 20 coin tosses Result Frequency Heads 12 Tails 8 Read 8 tails from the table.
Expected tails are .
8 observed; 10 expected. - A student only records heads and ignores tails. Explain why the record is incomplete.3 marks
Every valid trial must be recorded.
Ignoring results prevents a valid comparison of heads with all tosses.
You cannot check the total trials or the tails count. - Plan 40 draws from a bag of 2 red and 3 blue equal-size counters. Include selection, replacement, recording and expected reds.4 marks
Returning maintains red chance .
All recorded frequencies should total 40.
.
Draw without looking; record colour; replace and mix; repeat for 40 valid draws; expect 16 red.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum Mathematics
Year 7 single-stage sample spaces, equally likely outcomes, probabilities, expected frequencies, chance experiments, simulations and relative frequencies.
Official curriculum source ↗