Expected versus observed outcomes | Year 6

01 / LEARN

The idea, step by step

Expected means predicted, not promised

For repeated trials under an unchanged chance model, multiply the number of trials by the event probability to obtain an expected count. If green has probability one quarter, forty spins have an expected green count of ten.

An expected count is a prediction based on the model. Random variation means a particular set of forty spins can contain more or fewer than ten green results. That difference alone does not prove the spinner is unfair.

expected count = number of trials \times P (event)

Observed frequencies come from records

An observed frequency is the count actually recorded in the supplied results. Divide it by the number of trials to find the observed relative frequency. Check that all outcome counts add to the total number of trials.

These practice results contain fourteen greens in forty spins. The observed count differs from the prediction of ten. We can describe the difference and investigate further, but we should not alter the results to force agreement.

Supplied practice results: 40 spins of a fair four-colour spinner
Colour	Observed frequency
Red	8
Blue	9
Green	14
Yellow	9

A common trap

A fair coin must give exactly 25 heads in 50 tosses.

Instead: Twenty-five is the expected count. A fair coin can give a different count in a particular set of tosses.

02 / WORKED EXAMPLES

See the thinking

Example 1

Predict a green count

A fair spinner has four equally likely colours. Predict the number of greens in 40 spins.

The probability of green is $1 / 4$ .
A quarter of 40 is $40 \div 4 = 10$ .

10 greens expected; not guaranteed.

Example 2

Compare prediction and data

Using the supplied practice results, compare the predicted and observed green counts.

Supplied practice results: 40 spins of a fair four-colour spinner
Colour	Observed frequency
Red	8
Blue	9
Green	14
Yellow	9

The four frequencies add to $8 + 9 + 14 + 9 = 40$ .
Expected green count is $40 \times 1 / 4 = 10$ .
Observed green count is 14, which is 4 above expected.

Observed 14; expected 10; difference 4.

Check your understanding

A fair die is rolled 60 times. How many sixes are expected?

Show answer

10.

Each roll has probability $1 / 6$ of a six; $60 \div 6 = 10$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

How many heads are expected in 20 tosses of a fair coin?
2 marks
An event has probability $1 / 5$ . How many successes are expected in 50 trials?
2 marks
A fair coin gives 27 heads in 50 tosses. Find the observed percentage of heads.
2 marks
Using the supplied spinner results, how many fewer reds occurred than expected?
2 marks
Supplied practice results: 40 spins of a fair four-colour spinner
Colour Observed frequency
Red 8
Blue 9
Green 14
Yellow 9
A student changes the observed green frequency from 14 to 10 because the spinner should be fair. Explain the problem.
2 marks
A bag model gives a blue probability of 0.3. Predict blue counts in 100 trials, then explain whether 34 blue results proves the model wrong.
2 marks

Read the worked solutions

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

How many heads are expected in 20 tosses of a fair coin?
2 marks
$20 \times 1 / 2 = 10$ .
10.
An event has probability $1 / 5$ . How many successes are expected in 50 trials?
2 marks
$50 \div 5 = 10$ .
10.
A fair coin gives 27 heads in 50 tosses. Find the observed percentage of heads.
2 marks
$27 / 50 = 54 / 100 = 54 %$ .
54%.
Using the supplied spinner results, how many fewer reds occurred than expected?
2 marks
Supplied practice results: 40 spins of a fair four-colour spinner
Colour Observed frequency
Red 8
Blue 9
Green 14
Yellow 9
Expected reds: $40 \div 4 = 10$ .
Observed reds: 8; $10 - 8 = 2$ .
2 fewer.
A student changes the observed green frequency from 14 to 10 because the spinner should be fair. Explain the problem.
2 marks
Recorded results must be retained unless a recording error is established.
Fairness does not guarantee an exact expected count.
The student has replaced evidence with a prediction.
A bag model gives a blue probability of 0.3. Predict blue counts in 100 trials, then explain whether 34 blue results proves the model wrong.
2 marks
$100 \times 0.3 = 30$ .
A difference of four can occur through random variation; further evidence would be needed.
30 expected; 34 alone does not prove the model wrong.

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Curriculum reference & lesson scope

Australian Curriculum v9.0; SA Mathematics Curriculum 2026

Numerical likelihood; fractions/decimals/percentages; expected versus observed outcomes; larger simulations and variation.

Official curriculum source ↗

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