Theoretical and relative-frequency estimates | Year 9

01 / LEARN

Understand the idea.

Models and observations

A theoretical probability comes from a model and its assumptions, such as equally likely die faces. A relative frequency comes from observations: $\frac{successes}{trials}$ . These can differ because a finite experiment is subject to random variation.

To predict a count, multiply a suitable probability estimate by the number of future trials. This predicts an average over repeated sets of trials. It does not fix how many successes any one set must contain.

Compare like with like

A fair die model gives $P (6) = \frac{1}{6} \approx 0.167$ . Observing 18 sixes in 120 rolls gives $\frac{18}{120} = 0.15$ . Compare probabilities on the same scale; this difference alone does not establish that the die is unfair.

An estimate is useful for future trials only when conditions remain comparable. More correctly conducted trials usually reduce typical relative-frequency variation, but cannot fix biased recording or a simulation that follows different rules from the intended experiment.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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.

Worked solution

1
Estimate blue probability from the observed spins.
$\frac{42}{150} = 0.28$
Relative frequency divides the number of blue results by the total number of spins.
2
Scale the estimate to 200 similar spins.
$200 \times 0.28 = 56$
Using the same estimated blue rate requires comparable spinner conditions.
3
State the prediction with its limit.
$Estimated probability 0.28; about 56 blues$
The count is a prediction based on the observed rate, and random variation can change the actual count.

Answer

Estimated probability 0.28; about 56 blues.

EXAMPLE 02

Compare coins with a model

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

Worked solution

1
Find the theoretical probability of exactly one head.
$P (exactly one head) = \frac{2}{4} = 0.50$
HT and TH are the two favourable outcomes among HH, HT, TH and TT for fair independent tosses.
2
Calculate the observed relative frequency.
$\frac{44}{80} = 0.55$
Each trial consists of tossing both coins; 44 of the 80 trials produced exactly one head.
3
Compare the rates and the corresponding counts.
$\begin{aligned} 0.55 - 0.50 & = 0.05 \\ 80 \times 0.50 & = 40 \\ 44 - 40 & = 4 \end{aligned}$
The observed rate is 0.05 higher, corresponding to four more successes than the model's expected count.
4
Give a cautious conclusion.
Finite random variation can produce this difference; these results alone are insufficient to declare the coins unfair.

Answer

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

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

0.30

Relative frequency is successes divided by trials: $\frac{27}{90} = \frac{3}{10} = 0.30$ . It describes the observed proportion of successful trials.

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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 $\frac{1}{4}$ , 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
$\frac{16}{80} = 0.20$ .
$0.20$
An equal four-section spinner has one red section. Find $P (red)$ .
1 mark
One of four equally likely sections is red.
$\frac{1}{4}$
Using probability $\frac{1}{4}$ , predict the number of reds in 240 spins.
2 marks
$240 \times \frac{1}{4} = 60$ .
$60$ expected reds.
A game is won 21 times in 70 plays. Estimate the wins in another 150 similar plays.
2 marks
$\frac{21}{70} = 0.3$ .
$150 \times 0.3 = 45$ .
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: $200 + 300 = 500$ .
$\frac{144}{500} = 0.288$ ; 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 ↗

In this lesson

Understand the idea.

Models and observations

Compare like with like

See the method in action.

Predict a spinner count

Estimate blue probability from the observed spins.

Scale the estimate to 200 similar spins.

State the prediction with its limit.

Compare coins with a model

Find the theoretical probability of exactly one head.

Calculate the observed relative frequency.

Compare the rates and the corresponding counts.

Give a cautious conclusion.

Try it yourself.

Check your understanding

Keep practising