Complementary Events and Probability — AC9M8P01

• Identify the complement of an event within a stated sample space.
• Use P(A)+P(not A)=1 and P(not A)=1−P(A).
• Apply complements when outcomes are not equally likely.
• Interpret complement probabilities in practical and digital-classification contexts.

Recall sample space, event, probability from 0 to 1, fractions/decimals/percentages and the difference between a possible outcome and an event containing several outcomes.

The complement of event A is every outcome in the same sample space that is not in A. The two events cannot occur together and together cover the entire sample space, so P(A)+P(not A)=1.

The complement rule does not require equally likely outcomes. If a supermarket promotion gives a particular novelty toy with probability 0.18, then the probability of receiving any other toy is 1−0.18=0.82.

Always define the sample space before naming the complement. “Not late” is the complement of “late” only when those two categories cover every possibility being considered. In binary artificial-intelligence classification such as spam/not spam or fraud/not fraud, the model outputs may be treated as complementary categories only if the classification system is genuinely binary.

• “Opposite sounding” means complement. The two events must cover the whole stated sample space with no overlap.
• Complement only works for fair dice or coins. It works for any event whose probability is known.
• P(not A)=−P(A). Subtract from 1, not from zero.
• Two labels are automatically complementary. Check that no third category or unclassified outcome exists.

1. If P(bus is on time)=0.87, find P(bus is not on time).
2. For a fair die, find P(not rolling an even number).
3. Define the complement of “student chooses basketball” when the only choices are basketball or football.
4. Explain why “blue” is not necessarily the complement of “red” if several colours are possible.

1. If P(A)=0.42, find P(not A).
2. If P(not B)=0.09, find P(B).
3. A machine passes quality inspection with probability 0.965. Find the probability it does not pass.
4. A specific promotion prize has probability 7%. Find the probability of receiving another prize.
5. For a fair die, find P(not rolling 1 or 2).
6. Give a pair of events that are complementary and justify the sample space.
7. Give a pair of events that sound opposite but are not complementary because another outcome is possible.

An AI system reports “fraud 0.73” and “not fraud 0.19”. A student says these are complements because the labels are opposites. Explain what is wrong, what information is missing, and what the two reported probabilities would need to satisfy before the complement rule could be used directly.

1. What is a complementary event? Every outcome in the same sample space that is not in the original event.
2. Why do complementary probabilities sum to 1? Together they cover the whole sample space without overlap.
3. Do outcomes need to be equally likely? No; the rule uses the probability of the event, not equal-outcome counting.
4. What is the fastest complement calculation? P(not A)=1−P(A).

1. A weather service gives P(storm)=0.27. Find P(no storm) and explain why the answer is valid.
   Show that the two events exhaust the stated weather classification.
2. P(getting a particular toy)=0.14 in a promotion. Find P(not getting that toy) and explain why equal likelihood is irrelevant.
   Use the known event probability directly.
3. A system has categories “approved”, “rejected” and “manual review”. Explain why rejected is not the complement of approved.
   Identify the third outcome that would be omitted.

• I define the sample space first.
• I identify the true complement of an event.
• I use subtraction from 1 accurately.
• I can use complements in unequal-likelihood contexts.

Exit ticket: What two conditions must A and “not A” satisfy besides their probabilities summing to 1?

Australian Curriculum v9.0 — AC9M8P01: recognise that complementary events have a combined probability of one; use this relationship to calculate probabilities in applied contexts.

Victorian Curriculum F–10 Version 2.0 — Level 8, VC2M8P01: Exact.

NSW Mathematics K–10 Syllabus (2022) — Stage 4, MA4-PRO-C-01; MAO-WM-01: Partial. Stage 4 probability covers simple chance experiments; the national descriptor gives complementary-event reasoning a more explicit Year 8 emphasis.

• AC9M8P02 — representing two events
• AC9M8P03 — compound chance simulations

• Australian Curriculum v9.0
• Victorian Curriculum F–10 Version 2.0 Mathematics
• NSW Mathematics K–10 Syllabus (2022)

Curriculum equivalents for That complementary events have a combined probability of one; use...

Mapped skill: recognise that complementary events have a combined probability of one; use this relationship to calculate probabilities in applied contexts

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.

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.