Compound-event sample spaces | Year 9

01 / LEARN

The idea, step by step

Record the whole experiment

A compound experiment has at least two stages. For two coin tosses, HT means heads first and tails second. HT and TH are different ordered outcomes even though both have one head. Recording only the final stage loses information.

List systematically: keep the first result fixed while changing the second, then change the first. Two fair independent tosses give HH, HT, TH and TT.

Check equal chances before counting

For a fair coin and an independent equal three-section spinner, each of the six complete outcomes has probability 1/6. Count favourable entries and divide by six. The table helps check that none are missing.

The labels zero heads, one head and two heads are not equally likely. Exactly one head includes HT and TH, while the other labels each include just one ordered outcome.

Fair coin followed by equal spinner 1, 2, 3
Coin / spinner	1	2	3
H	H1	H2	H3
T	T1	T2	T3

A common trap

Three possible head-count labels must each have probability 1/3.

Instead: Count the equally likely ordered outcomes underneath each label. One head has two outcomes.

02 / WORKED EXAMPLES

See the thinking

Example 1

Two fair coins

Find the probability of exactly one head in two fair independent tosses.

List HH, HT, TH, TT.
HT and TH are favourable: two of four.

2 / 4 = 1 / 2

Example 2

Coin and spinner

For the displayed coin-spinner experiment, find the probability of a tail and an odd number.

Fair coin followed by equal spinner 1, 2, 3
Coin / spinner	1	2	3
H	H1	H2	H3
T	T1	T2	T3

There are six equally likely outcomes.
T1 and T3 meet both conditions.

2 / 6 = 1 / 3

Check your understanding

What is P(no heads) in two fair independent tosses?

Show answer

$1 / 4$

Only TT is favourable among four ordered outcomes.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

List the sample space for two fair independent coin tosses.
1 mark
Find P(two heads) for two fair independent coin tosses.
2 marks
A fair coin and an independent fair die are used. How many ordered outcomes are possible?
2 marks
For a fair coin and independent fair die, find P(heads and die greater than 4).
2 marks
An equal spinner labelled A, B, C is spun twice independently. Find P(same letter).
2 marks
Explain why zero, one and two heads are not equally likely in two fair independent tosses.
3 marks

Read the worked solutions

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

List the sample space for two fair independent coin tosses.
1 mark
Keep the first result H for two entries, then T.
HH, HT, TH, TT.
Find P(two heads) for two fair independent coin tosses.
2 marks
Only HH is favourable out of four.
$1 / 4$
A fair coin and an independent fair die are used. How many ordered outcomes are possible?
2 marks
2 coin results times 6 die results gives 12.
12
For a fair coin and independent fair die, find P(heads and die greater than 4).
2 marks
H5 and H6 are favourable among 12 outcomes: 2/12=1/6.
$1 / 6$
An equal spinner labelled A, B, C is spun twice independently. Find P(same letter).
2 marks
AA, BB and CC are favourable among nine outcomes.
$1 / 3$
Explain why zero, one and two heads are not equally likely in two fair independent tosses.
3 marks
Group HH, HT, TH, TT by head count.
Their probabilities are 1/4, 1/2 and 1/4.
Exactly one head has two ordered outcomes; zero and two heads have one each.

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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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