The idea, step by step
Translate the event first
“And” requires both conditions. “Or” normally includes either condition or both; “exactly one” excludes the overlap. “At least one” excludes only the outcome where neither occurs. For a die, even or greater than 4 is {2,4,5,6}; count 6 only once.
Combine paths carefully
Multiply stage probabilities along a complete path when stages are independent, or when the changing branch probabilities are specified. Add mutually exclusive complete outcomes. Without replacement, the remaining bag changes: do not automatically multiply the original probabilities. A physical-outcome list or labelled tree is often the clearest method.
A common trap
Or always means exactly one condition.
Instead: Unless explicitly stated otherwise, or includes the overlap.
See the thinking
Worked example 1
Find P(at least one head) for two independent fair coins.
- Outcomes are HH,HT,TH,TT.
- Three of four have at least one head.
- Equivalently exclude TT:1−1/4.
Worked example 2
A bag has B1, B2 and G. Draw two tokens at random without replacement. Find P(one of each colour).
- Six ordered physical pairs are possible.
- Four are B1G,B2G,GB1,GB2.
- 4/6=2/3.
Check your understanding
Find P(odd or multiple of 3) for a fair die.
Show answer
2/3.
The union is {1,3,5,6}, so 4/6=2/3.
Your turn
6 questions · 14 marks · Work at your own pace.
Read the questions online
- A fair coin and standard die are independent. Find P(head and die greater than 4).2 marks
- A bag has 2 red and 3 blue tokens. Draw at random and replace the token before a second random draw. Find P(two red).2 marks
- Find P(two heads or two tails) for two independent fair coins.2 marks
- Two independent fair spinners each show 1,2,3. Find P(sum 4).3 marks
- Find P(even or greater than 4) for a fair standard die.2 marks
- A bag has B1, B2 and G. Draw two tokens at random without replacement. Find P(both blue).3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A fair coin and standard die are independent. Find P(head and die greater than 4).2 marks
1/2×2/6=1/6.
1/6. - A bag has 2 red and 3 blue tokens. Draw at random and replace the token before a second random draw. Find P(two red).2 marks
2/5×2/5=4/25.
4/25. - Find P(two heads or two tails) for two independent fair coins.2 marks
HH or TT gives 2 of 4 outcomes.
1/2. - Two independent fair spinners each show 1,2,3. Find P(sum 4).3 marks
Favourable pairs (1,3),(2,2),(3,1) give 3 of 9.
1/3. - Find P(even or greater than 4) for a fair standard die.2 marks
Outcomes {2,4,5,6} give 4/6.
2/3. - A bag has B1, B2 and G. Draw two tokens at random without replacement. Find P(both blue).3 marks
Favourable pairs B1B2 and B2B1 give 2 of 6.
1/3.
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 ↗