Expected frequencies | Year 7

01 / LEARN

The idea, step by step

From chance to predicted count

For event probability $p$ , the expected frequency in $n$ repetitions is $n p$ . If a quarter of an ideal spinner is blue, about a quarter of many spins should be blue. For 80 spins, $80 \times 1 / 4 = 20$ blue results are expected.

The model probability must apply throughout the repeated experiment. Replacing a drawn counter can preserve the bag’s colour chances. The prediction does not mean that every group of four spins contains one blue result.

expected frequency = trials \times probability

Expected is not guaranteed

Random variation means actual counts may be above or below their expectation. A fair coin can produce more heads than tails in a short run. Larger runs often make proportions more stable, but the raw difference in counts need not decrease.

An expected count can be fractional. Six spins with blue chance $1 / 4$ give expected count 1.5. Each individual run has a whole blue count; 1.5 describes an average across many comparable runs, not half of a recorded outcome.

A common trap

The event must occur soon to make up for earlier misses.

Instead: An expected count does not force a sequence or a catch-up outcome. Future trials still follow the stated chance model.

02 / WORKED EXAMPLES

See the thinking

Example 1

Predict multiples on a die

A fair die is rolled 120 times. How many multiples of 3 are expected?

The favourable faces are 3 and 6: chance $2 / 6 = 1 / 3$ .
Multiply $120 \times 1 / 3 = 40$ .
State this as a prediction rather than a required exact count.

40 results are expected.

Example 2

A prediction between whole numbers

An ideal spinner has blue chance $1 / 4$ . Interpret the expected blue count in 6 spins.

Calculate $6 \times 1 / 4 = 1.5$ .
An individual run only has a whole number of blue results.
Across many six-spin runs, their average count is expected to be near 1.5.

Expected count 1.5; it is an average prediction, not a possible half-result.

Check your understanding

A fair coin is tossed 50 times. How many heads are expected?

Show answer

25 heads.

$50 \times 1 / 2 = 25$ ; actual counts need not be exactly 25.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

An event has probability $1 / 5$ . Find the expected frequency in 40 trials.
2 marks
An ideal spinner is half red. Find the expected red count in 90 spins.
2 marks
A bag has 3 green and 2 white equal-size counters. Each randomly chosen counter is returned and remixed. How many green results are expected in 50 draws?
3 marks
A fair die is rolled 60 times. How many results greater than 4 are expected?
2 marks
An event has chance $1 / 4$ and is repeated 10 times. Is an expected frequency of 2.5 a mistake? Explain.
2 marks
A fair coin gave 31 heads in 50 tosses instead of the expected 25. Does that alone show the prediction calculation was wrong?
3 marks

Read the worked solutions

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

An event has probability $1 / 5$ . Find the expected frequency in 40 trials.
2 marks
$40 \times 1 / 5 = 8$ .
8.
An ideal spinner is half red. Find the expected red count in 90 spins.
2 marks
Red chance is $1 / 2$ .
$90 \times 1 / 2 = 45$ .
45.
A bag has 3 green and 2 white equal-size counters. Each randomly chosen counter is returned and remixed. How many green results are expected in 50 draws?
3 marks
Returning maintains chance $3 / 5$ .
$50 \times 3 / 5 = 30$ .
30.
A fair die is rolled 60 times. How many results greater than 4 are expected?
2 marks
The favourable faces are 5 and 6: chance $1 / 3$ .
$60 \times 1 / 3 = 20$ .
20.
An event has chance $1 / 4$ and is repeated 10 times. Is an expected frequency of 2.5 a mistake? Explain.
2 marks
$10 \times 1 / 4 = 2.5$ .
A predicted average count can be fractional although actual counts are whole.
No.
A fair coin gave 31 heads in 50 tosses instead of the expected 25. Does that alone show the prediction calculation was wrong?
3 marks
The expected count remains $50 \times 1 / 2 = 25$ .
31 is one random observed count; expected does not mean guaranteed.
No.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Year 7 single-stage sample spaces, equally likely outcomes, probabilities, expected frequencies, chance experiments, simulations and relative frequencies.

Official curriculum source ↗

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