The idea, step by step
Observed versus predicted
Relative frequency is successes divided by total trials. Theoretical probability comes from the stated model, such as a fair coin or equal spinner sectors. Expected frequency equals trial count multiplied by model probability. It describes an average across repeated experiments, not a promised result in one batch.
Variation is normal
A fair coin need not give exactly half heads in each batch. Larger repeated trials often stabilise proportions, but the estimate need not improve at every step. Differences prompt checks of assumptions, recording and sample size; they do not alone prove bias. Combine batches using total successes over total trials, not an unweighted mean of percentages.
A common trap
Expected frequency must equal observed frequency.
Instead: Expectation describes the model average, not a guaranteed batch count.
See the thinking
Worked example 1
A coin gives 40 heads in 100 tosses. Compare its relative frequency with the fair-coin model.
- Observed=40/100=0.4.
- Theory=0.5; expected heads=50.
- Finite experiments can differ from expectation.
Worked example 2
A model gives P(head and die 6)=1/12. There are 8 successes in 120 trials.
- Expected=120×1/12=10.
- Experimental=8/120=1/15.
- 8 is not required to equal 10.
Check your understanding
For P(win)=0.3, find expected wins in 200 trials.
Show answer
60.
200×0.3=60.
Your turn
6 questions · 14 marks · Work at your own pace.
Read the questions online
- An event occurs 18 times in 30 trials. Find relative frequency.2 marks
- For P(win)=1/4, find expected wins in 240 trials.2 marks
- A fair die gives 51 sixes in 240 rolls. State experimental and theoretical probabilities.3 marks
- There are 23 successes in 80 trials, then 37 in 120 more. Find combined relative frequency.3 marks
- Using an estimated probability 0.38, predict successes in 300 trials.2 marks
- Why are exactly 10 heads not guaranteed in 20 fair tosses?2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- An event occurs 18 times in 30 trials. Find relative frequency.2 marks
18÷30=0.6.
0.6. - For P(win)=1/4, find expected wins in 240 trials.2 marks
240×1/4=60.
60. - A fair die gives 51 sixes in 240 rolls. State experimental and theoretical probabilities.3 marks
Actual counts define the estimate; the fair model defines theory.
51/240=17/80; theoretical 1/6. - There are 23 successes in 80 trials, then 37 in 120 more. Find combined relative frequency.3 marks
(23+37)/(80+120)=60/200=0.3.
0.3. - Using an estimated probability 0.38, predict successes in 300 trials.2 marks
300×0.38=114; it remains a prediction.
114. - Why are exactly 10 heads not guaranteed in 20 fair tosses?2 marks
A long-run model proportion does not fix one batch’s outcome.
Many counts remain possible despite an expected count of 10.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum
Complementary events; two-stage sample spaces; tables, trees and Venn diagrams; probabilities of combined outcomes; experimental versus theoretical results; compound-event simulations and describing results. Formal conditional probability is Year 10.
Official curriculum source ↗