Notation
P(A|B)=P(A∩B)/P(B), if P(B)>0.
Learning goals: what you will learn
- explain the central idea: Use conditional-probability language precisely and identify how the relevant sample space changes when information is given.
- choose and apply an appropriate method without relying on keyword matching
- check results using units, substitution, estimation, a second representation or contextual reasonableness
- justify a conclusion and communicate limitations where the context requires them
Prerequisite knowledge
Revise fractions, probability, two-way tables, Venn-style set thinking and basic event notation.
Concept teaching
P(A|B) means “probability of A given B”. Once B is known, only outcomes inside B remain possible. Independent events satisfy P(A|B)=P(A); dependent events do not.
Language
“of B”, “given B”, “knowing B” → denominator is B group/sample space.
Independent
P(A|B)=P(A).
Tree
Multiply along branches; branch probabilities may change after a dependent outcome.
Worked examples
1. Two-way table
Of 60 seniors, 24 prefer option A. P(A|senior)=24/60=0.4.
2. Reverse condition
If 24 seniors prefer A out of 54 total A-preferrers, P(senior|A)=24/54≈0.444. This is not same denominator as Example 1.
3. Venn interpretation
If |B|=40 and |A∩B|=10, then P(A|B)=10/40=0.25.
4. Independent coin tosses
P(second head | first head)=1/2, same as P(second head); tosses are independent.
5. Without replacement
Bag 3 red,2 blue. P(second red | first red)=2/4=1/2; condition changes composition.
6. With replacement
Replace first red: P(second red | first red)=3/5, unchanged from initial probability.
7. Tree path
Without replacement, P(red then blue)=3/5×2/4=3/10.
8. “If…then” logic caution
“If selected student is senior, probability of A is 40%” conditions on senior. It does not mean “40% of A-preferrers are seniors”.
9. Recommendation system
A system can estimate P(click | user viewed category X). The condition narrows to users/events where X was viewed.
10. Text generation intuition
A language model assigns probabilities to next tokens conditional on preceding context; the context changes the probability distribution of possible next tokens.
Common misconceptions and corrections
- Swapping P(A|B) and P(B|A).
- Using total sample as denominator after a condition is given.
- Assuming events are independent without checking.
- Keeping branch probabilities unchanged in sampling without replacement.
- Reading “if…then” as ordinary implication without identifying the restricted probability group.
Guided practice
A table shows 30 bus users, of whom 12 are late, and 70 non-bus users, of whom 14 are late. Find P(late | bus) and P(bus | late). Explain why the two denominators differ.
Teacher check: require a written method choice and one verification step before revealing the worked solution.
Independent practice
- 30 of 50 juniors choose X. Find P(X|junior).
- 20 of 40 X-choosers are seniors. Find P(senior|X).
- If |B|=25 and |A∩B|=5, find P(A|B).
- For fair coin, find P(H₂|H₁).
- Bag 4R,1B without replacement: P(R₂|R₁)?
- Same bag with replacement?
- Find P(R then B) without replacement.
- Explain why P(A|B) may differ from P(A).
- State independence condition using probabilities.
- Describe condition in P(click|viewed sports).
Check answers and explanations
- 0.6.
- 0.5.
- 0.2.
- 0.5.
- After red, 3R/4 total → 3/4.
- 4/5.
- 4/5×1/4=1/5.
- Knowing B changes/restricts the sample space when events are dependent.
- P(A|B)=P(A), for P(B)>0.
- Only cases where sports was viewed.
Reasoning and problem-solving task
A school reports: '80% of students who achieved an A used the revision platform.' A parent interprets this as '80% of platform users get an A.' Explain the conditional-probability error and construct a numerical example where the first statement is true but the parent's statement is far from true.
Important questions and answers
- 30 of 50 juniors choose X. Find P(X|junior).
- 0.6.
- 20 of 40 X-choosers are seniors. Find P(senior|X).
- 0.5.
- If |B|=25 and |A∩B|=5, find P(A|B).
- 0.2.
Assessment-style questions
In a cohort, 45 students study Chemistry, 30 of those study Physics, and 55 of the 75 non-Chemistry students study Physics. Calculate P(Physics | Chemistry) and P(Chemistry | Physics), then explain why reversing a condition changes the probability. [6 marks]
Marking focus: method selection, mathematically correct working, interpretation and justification.
Review hints
- circle the condition
- rebuild the denominator from the conditioned group
- do not reverse the words
- test converses with counterexamples
Exit ticket: mastery check
- I can explain the concept without copying a formula sheet.
- I can solve a routine example and check the result.
- I can choose a method in an unfamiliar problem.
- I can explain a common error and correct it.
- I can connect the answer back to the context, including units or limitations.
Teacher and parent guidance
For teachers
Read conditional statements aloud and have students physically cover rows/columns outside the conditioned group before calculating. This makes the denominator change visible.
For parents and carers
Ask 'Who are we talking about now?' whenever a question says 'given' or 'of'. If your child names the restricted group before calculating, they are likely on track.
Curriculum alignment
Australian Curriculum: AC9M10P01 — Year 10 Probability. Use conditional-probability language precisely and identify how the relevant sample space changes when information is given.
Victoria: VC2M10P01 — Level 10 Probability
NSW: Stage 5 Core — Probability; Path — Further probability
Alignment explanation: The explicit teaching and worked examples address the Australian Curriculum concept directly. The Victorian mapping follows the current Version 2.0 descriptor structure; where Victoria combines or extends content, that difference is stated rather than hidden. NSW uses a Stage 5 Core–Paths structure, so this page maps to the relevant content group(s) and Working mathematically processes instead of inventing a Year 10 one-to-one code.
| Lesson component | Australian Curriculum | Victoria | NSW |
|---|---|---|---|
| Explicit concept teaching and worked examples | AC9M10P01 | VC2M10P01 — Level 10 Probability | Stage 5 Core — Probability; Path — Further probability |
| Guided and independent practice | Builds fluency and application for AC9M10P01 | Practises the mapped Level 10/10A knowledge as applicable | Practises the mapped Stage 5 Core/Path content |
| Reasoning and assessment tasks | Applies reasoning/problem solving in the descriptor context | Supports Victorian reasoning and modelling expectations | Embeds Working mathematically: reasoning, problem solving and communication |
Australian Curriculum elaborations
AC9M10P01: use language of “if ... then”, “given”, “of”, “knowing that” to describe and interpret conditional probability.
- E1: use two-way tables/Venn diagrams to interpret conditional statements and common language errors. Examples 1–3,8.
- E2: use arrays/tree diagrams to represent and compare dependent/independent events. Examples 4–7.
- E3: investigate conditional probability in NLP tasks such as generation, translation, augmentation and recommendations. Examples 9–10.
Practice and teaching resources
Official curriculum references
- Australian Curriculum Version 9 — ACARA
- Victorian Curriculum Mathematics Version 2.0 — VCAA
- NSW Mathematics K–10 Syllabus (2022) — NSW Curriculum/NESA
Official wording is paraphrased on SkillrHub; use the linked curriculum sites as the source of record.
Other curriculum comparisons retained
US closest CCSS HSS-CP.A.3–5; UK GCSE conditional probability/tree diagrams; NSW Stage 5 and Victorian Level 10 probability; comparable Canadian/NZ secondary conditional probability.
🎥 Optional Video Lesson
The SkillrHub lesson remains the primary learning resource. This optional video reinforces the explanation; you can complete the lesson and practice without watching.
Before you watch:
- Pause after each worked example.
- Try the examples yourself.
- Return to the SkillrHub lesson before continuing.
Recommended: Calculating conditional probability
Khan Academy — Restricting the sample space after learning that an event has occurred.
As you watch: What changes in the denominator when a condition is given?
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Curriculum equivalents: Victoria, NSW and international
Curriculum equivalents for The language of “if ... then”, “given”, “of”, “knowing that”...
Mapped skill: use the language of “if ... then”, “given”, “of”, “knowing that” to describe and interpret situations involving conditional probability
These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.
| Region | Curriculum framework | Closest level or code |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M10P01 · Year 10 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M10P01 · Level 10 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA5-PRO-P-01 · Stage 5 |
| United States (USA) | Common Core State Standards for Mathematics | Grades 9–10 band |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 10 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 11, Key Stage 4 |
| India | NCERT / CBSE — Mathematics | Class 10 |
Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.
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Topic reference: AC9M10P01 — Conditional Probability Language — AC9M10P01
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