Theoretical and relative-frequency estimates | Year 9

01 / LEARN

The idea, step by step

Models and observations

Theoretical probability follows from assumptions such as fair, independent coin tosses. Relative frequency comes from data: number of successes divided by number of trials. Random variation can make the observed frequency differ from a model even when the model is suitable.

To predict a count, multiply probability by number of trials. The result is an expected count over repeated sets, not a fixed count for each set.

Compare like with like

A fair die predicts a six on 1/6 of rolls. If 18 sixes occur in 120 rolls, the relative frequency is 0.15, slightly below 1/6. That difference alone does not prove the die is unfair.

Report how many trials were made and whether conditions remained similar. A larger well-designed sample usually gives a more stable estimate, but cannot repair a model that uses the wrong rules.

A common trap

Probability 0.3 guarantees 30 successes in every 100 trials.

Instead: It predicts an average count; individual sets vary.

02 / WORKED EXAMPLES

See the thinking

Example 1

Predict a spinner count

A spinner gives blue 42 times in 150 spins. Estimate P(blue), then predict the count in another 200 similar spins.

Relative frequency is $42 / 150 = 0.28$ .
Predicted count is $200 \times 0.28 = 56$ .

Estimated probability 0.28; about 56 blues.

Example 2

Compare coins with a model

Two fair independent coins are tossed 80 times. Exactly one head appears 44 times. Compare observation and theory.

The sample space HH, HT, TH, TT gives theoretical probability 1/2.
Observed frequency is $44 / 80 = 0.55$ .
A difference of 0.05 may occur through random variation.

Observed 0.55 versus theoretical 0.50; not enough alone to declare unfair coins.

Check your understanding

An event happens 27 times in 90 trials. Find its relative frequency.

Show answer

0.30

$27 / 90 = 0.30$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

An event occurs 16 times in 80 trials. Find its relative frequency.
1 mark
An equal four-section spinner has one red section. Find P(red).
1 mark
Using probability 1/4, predict the number of reds in 240 spins.
2 marks
A game is won 21 times in 70 plays. Estimate the wins in another 150 similar plays.
2 marks
A fair coin gives 14 heads in 20 tosses. Does this prove bias? Explain.
3 marks
A spinner gives green 60 times in 200 spins and 84 times in another 300. Find the combined relative frequency.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

An event occurs 16 times in 80 trials. Find its relative frequency.
1 mark
$16 / 80 = 0.20$ .
0.20
An equal four-section spinner has one red section. Find P(red).
1 mark
One of four equally likely sections is red.
$1 / 4$
Using probability 1/4, predict the number of reds in 240 spins.
2 marks
$240 \times 1 / 4 = 60$ .
60 expected reds.
A game is won 21 times in 70 plays. Estimate the wins in another 150 similar plays.
2 marks
$21 / 70 = 0.3$ .
$150 \times 0.3 = 45$ .
About 45 wins.
A fair coin gives 14 heads in 20 tosses. Does this prove bias? Explain.
3 marks
The expected head count is 10, not a guaranteed 10.
More controlled trials would help assess the model.
No; an uneven short run can occur by chance.
A spinner gives green 60 times in 200 spins and 84 times in another 300. Find the combined relative frequency.
3 marks
Successes: $60 + 84 = 144$ .
Trials: $200 + 300 = 500$ .
$144 / 500 = 0.288$ ; do not average the rates without weighting.
0.288

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Year 9 compound-event sample spaces, probabilities, experiments and simulations.

Official curriculum source ↗

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