The idea, step by step
State what the evidence describes
A result of 32 successes in 100 trials gives relative frequency 0.32. Applying this to future trials assumes similar conditions. It does not guarantee the next outcome or an exact count in the next hundred.
Before comparing experiments, check the event definition, rules, equipment, sample size and recording method. A large sample from the wrong experiment can still answer the wrong question.
Give a conclusion and its limit
A useful conclusion gives an estimate, compares it with a model and states a limitation. For example, 0.32 is near 1/3, so the difference may be random variation; repeat with more correctly conducted trials before making a stronger claim.
After a losing streak, a success is not due if trials are independent and probabilities unchanged. Also, a short run matching a fair-model prediction does not prove a device fair.
A common trap
A long losing streak increases the next-win chance.
Instead: With independent trials and unchanged rules, the next probability stays the same.
See the thinking
Unequal sample sizes
Group A has 8 successes in 20 trials; B has 54 in 100. Compare rates and combine them.
- A: ; B: .
- Combined: .
- The unweighted mean 0.47 gives equal weight to unequal samples and is unsuitable here.
A losing streak
A fair independent coin gives T five times. Sam says H is more likely next. Assess this.
- Past outcomes do not change the two equally likely results of the next toss.
- Independence keeps P(H) at 1/2.
Check your understanding
A model predicts 25 successes per 100 trials on average. A sample gives 23. Must the model be wrong?
Show answer
No.
The expected count is not fixed, so small differences can arise by chance.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- A game has win probability 0.2. After four losses, what is the chance of the next independent win?1 mark
- A device gives 48 successes in 150 trials. State the observed relative frequency.1 mark
- Two reports give 18 and 25 successes but no trial totals. What is missing for a fair rate comparison?2 marks
- Two groups report 12 successes in 40 trials and 28 in 60. Find the combined estimate.2 marks
- A simulation replaces counters, but the real game keeps the first out. Can its result be used unchanged?3 marks
- A model probability is 1/3. An experiment gives 31 successes in 90 trials. Write a conclusion with a limitation.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A game has win probability 0.2. After four losses, what is the chance of the next independent win?1 mark
The unchanged independent trial has the same chance.
0.2 - A device gives 48 successes in 150 trials. State the observed relative frequency.1 mark
.
0.32 - Two reports give 18 and 25 successes but no trial totals. What is missing for a fair rate comparison?2 marks
Counts alone do not show proportions.
The number of trials in each experiment. - Two groups report 12 successes in 40 trials and 28 in 60. Find the combined estimate.2 marks
successes in trials.
0.40 - A simulation replaces counters, but the real game keeps the first out. Can its result be used unchanged?3 marks
Without replacement, the remaining counts change.
Update the collection within each trial.
No; it models different second-draw probabilities. - A model probability is 1/3. An experiment gives 31 successes in 90 trials. Write a conclusion with a limitation.3 marks
.
Compare with 0.333... and recommend more correctly conducted trials rather than claiming proof.
The estimate is about 0.344, close to 1/3; finite-sample variation remains.
Curriculum reference & lesson scope
Australian Curriculum v9 / SA Curriculum Mathematics
Year 9 compound-event sample spaces, probabilities, experiments and simulations.
Official curriculum source ↗