Two-stage sample spaces | Year 8

01 / LEARN

Understand the idea.

Keep the stages in order

A two-stage outcome records what happens first and second, keeping their order. A head followed by spinner 2 can be written H2. List every second-stage possibility beneath each first-stage possibility so that no complete result is omitted.

A table uses first-stage results for rows and second-stage results for columns; each cell is a complete ordered outcome. A tree shows the same construction through branches. Either method lets you check the systematic list rather than relying on guesses.

Count elementary outcomes

If every first-stage outcome allows the same number of second-stage possibilities, multiply the counts. Two six-sided dice produce $6 \times 6 = 36$ ordered pairs. To turn a count into a probability by division, also justify that the elementary outcomes are equally likely.

Do not replace elementary outcomes by summaries too early. Two dice can show 7 in six different ordered ways but 2 in only one. Similarly, two differently labelled blue tokens remain distinct possible draws even when their colours match.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

List outcomes for a fair coin then an independent fair spinner labelled 1,2,3.

Worked solution

1
List outcomes beginning with a head.
$H 1, H 2, H 3$
After H, each of the spinner's three labels is possible.
2
List outcomes beginning with a tail.
$T 1, T 2, T 3$
The same three spinner labels are possible after T.
3
Check the complete count and likelihoods.
$2 \times 3 = 6, \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$
The fair coin and independent fair spinner make all six listed ordered outcomes equally likely.

Answer

H1,H2,H3,T1,T2,T3.

EXAMPLE 02

Worked example 2

Is “one head” a single elementary outcome for two ordered fair coins?

Worked solution

1
List the ordered coin results.
$H H, H T, T H, T T$
First-coin and second-coin results distinguish HT from TH.
2
Select outcomes containing exactly one head.
${H T, T H}$
A head can occur on either the first or the second coin, so this event contains two elementary outcomes.
3
Calculate the event probability.
$P (one head) = \frac{2}{4} = \frac{1}{2}$
Two favourable results occur among four equally likely ordered outcomes of fair coin tosses.

Answer

No; P(one head)=1/2.

03 / YOUR TURN

Try it yourself.

Check your understanding

How many ordered outcomes come from two independent four-sector spinners?

Show answer & reasoning

16.

For each of four first-spinner results, there are four second-spinner results. Hence $4 \times 4 = 16$ ordered outcomes; counting them does not require assuming that their probabilities are equal.

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

List the sample space for two ordered coin tosses.
2 marks
A coin then an A,B,C spinner has how many ordered outcomes?
1 mark
List outcomes for two ordered spinners each labelled 1,2.
2 marks
For a coin then a 1,2,3 spinner, list outcomes with a tail or spinner 3.
3 marks
How many ordered pairs can two standard dice produce? Why do $(2, 5)$ and $(5, 2)$ differ?
2 marks
A bag holds R1,R2,B. Draw two without replacement. List the ordered physical pairs.
3 marks

Read the worked solutions

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

List the sample space for two ordered coin tosses.
2 marks
Pair each first result with both second results.
HH,HT,TH,TT.
A coin then an A,B,C spinner has how many ordered outcomes?
1 mark
$2 \times 3 = 6$ .
6.
List outcomes for two ordered spinners each labelled 1,2.
2 marks
Both stages are recorded.
$(1, 1)$ , $(1, 2)$ , $(2, 1)$ , $(2, 2)$ .
For a coin then a 1,2,3 spinner, list outcomes with a tail or spinner 3.
3 marks
Include the overlapping T3 only once.
T1,T2,T3,H3.
How many ordered pairs can two standard dice produce? Why do $(2, 5)$ and $(5, 2)$ differ?
2 marks
$6 \times 6 = 36$ , and die order is retained.
36; the first-die result differs.
A bag holds R1,R2,B. Draw two without replacement. List the ordered physical pairs.
3 marks
There are 3 first choices and 2 remaining choices, giving 6 pairs.
(R1,R2),(R1,B),(R2,R1),(R2,B),(B,R1),(B,R2).

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum

Complementary events; two-stage sample spaces; tables, trees and Venn diagrams; probabilities of combined outcomes; experimental versus theoretical results; compound-event simulations and describing results. Formal conditional probability is Year 10.

Official curriculum source ↗

In this lesson

Understand the idea.

Keep the stages in order

Count elementary outcomes

See the method in action.

Worked example 1

List outcomes beginning with a head.

List outcomes beginning with a tail.

Check the complete count and likelihoods.

Worked example 2

List the ordered coin results.

Select outcomes containing exactly one head.

Calculate the event probability.

Try it yourself.

Check your understanding

Keep practising