The idea, step by step
Keep the stages in order
A two-stage outcome records both what happens first and what happens second. H2 can mean a head followed by spinner result 2. Tables, systematic lists and trees help avoid omissions. Begin with each first-stage possibility and attach every allowed second-stage possibility.
Count elementary outcomes
When every first-stage outcome permits the same number of second-stage outcomes, multiply the counts. Two standard dice have 6×6=36 ordered number pairs. Their sums are not equally likely: several pairs may produce one sum. Likewise, colour labels can conceal several distinct physical token pairs.
A common trap
Two dice have 11 equally likely outcomes because sums run from 2 to 12.
Instead: The 36 ordered pairs are equally likely; the sums have different numbers of pairs.
See the thinking
Worked example 1
List outcomes for a fair coin then an independent fair spinner labelled 1,2,3.
- With H: H1,H2,H3.
- With T: T1,T2,T3.
- 2×3=6 equally likely outcomes.
Worked example 2
Is “one head” a single elementary outcome for two ordered fair coins?
- Sample space: HH,HT,TH,TT.
- One head includes HT and TH.
- Two of four outcomes are favourable.
Check your understanding
How many ordered outcomes come from two independent four-sector spinners?
Show answer
16.
4×4=16.
Your turn
6 questions · 13 marks · Work at your own pace.
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
2×3=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
6×6=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).
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 ↗