Events and Complements

Overview

In this third lesson of Probability: From Chance to Models, students move from identifying equally likely outcomes to describing and calculating probabilities of events. They interpret common probability language, use outcome lists or grids, and introduce complementary probability as an efficient method for finding “not” events.

Learning intentions

• WALT list the relevant outcomes for an event accurately.
• WALT interpret probability language such as odd, prime, multiple, greater than, less than, at least, and inclusive.
• WALT calculate the probability of an event and its complement using (P(\text{not }A)=1-P(A)).
• WALT communicate probability reasoning using fractions, decimals, and appropriate mathematical language.

Success criteria

• I can list the outcomes that belong to an event.
• I can explain what words such as “at least” and “inclusive” mean in context.
• I can calculate a complement using (P(\text{not }A)=1-P(A)).
• I can explain why my answer is reasonable and give it as a simplified fraction or decimal.

Curriculum links

• Interpret and apply mathematical and statistical information in context by making informed decisions from mathematical information.
• Demonstrate mathematical reasoning by using mathematical methods, concepts, terms, and appropriate representations.
• Use mathematical methods to explore problems connected with life in Aotearoa New Zealand or the Pacific, communicating accurate mathematical information.
• Te Mātaiaho: Mathematics and Statistics learning area — use mathematical thinking, representations, reasoning, and communication to make sense of chance in meaningful contexts.

Lesson structure (95 minutes)

1. 0–8 min · Hook and retrieval. Open with the opening chance challenge and display: “A fair six-sided die is rolled. Is rolling a number greater than 4 more likely than rolling an even number?” Students make an individual estimate, justify it briefly, then compare with a partner. Teacher revisits sample space, equally likely outcomes, and probability as favourable outcomes over total outcomes.

2. 8–23 min · Explicit teaching: events and language. Use the event-language teaching slides to model event notation and outcome lists for a fair die: odd, even, prime, multiple of 3, greater than 4, and less than 4. Explicitly teach that “at least 4” means 4 or more, “at most 4” means 4 or less, and an inclusive range includes both endpoints. Students annotate examples and sort each event into an accurate outcome list. Check understanding with mini-whiteboards: “List the outcomes for numbers from 2 to 5 inclusive.”

3. 23–43 min · Guided practice with representations. Distribute the dyslexia-friendly events and outcomes worksheet. Teacher models the first two questions using an outcome grid and, where helpful, a physical die or spinner demonstration. Students complete questions involving one die and two-digit number cards, identifying events such as odd, prime, a multiple, and an inclusive range. Pause after each section for students to compare lists and explain how they interpreted the wording.

4. 43–55 min · Introducing complements. Return to the complement model and worked examples. Teacher shades an outcome grid to show event (A) and “not (A)”, then develops (P(\text{not }A)=1-P(A)). Model: for a die, if (A) is “roll a prime”, (P(A)=3/6), so (P(\text{not }A)=1-3/6=3/6=1/2). Students predict and verify complements for “roll an even number”, “roll greater than 4”, and “roll at least 3”. Emphasise that an event and its complement contain every possible outcome exactly once.

5. 55–78 min · Paired problem-solving and reasoning. Students complete the remaining worksheet questions in pairs, including direct and complementary methods. Problems use fair dice, numbered cards, and a simple Aotearoa context such as selecting a number from a list of bus routes or jersey numbers. Students must underline key words, list the sample space, identify the event, calculate, and write one sentence explaining their method. Teacher circulates, checking that students distinguish “greater than” from “greater than or equal to” and that complements are taken from the correct sample space. Use the paired-task instructions and discussion prompts for pacing and peer-check questions.

6. 78–88 min · Extension and class discussion. Advanced learners complete the multi-condition LOTTO challenge: determine the probability of selecting a number from 1–40 that is a multiple of 3 or a prime, then solve a related “not a multiple of 3 or prime” question using both direct listing and a complementary method. Students explain overlaps carefully and compare which method is more efficient. Other students select one completed problem to present using an outcome grid or annotated list.

7. 88–95 min · Plenary and exit check. Use the final reflection and exit questions. Students answer independently: “A number is chosen at random from 1–10. Find the probability of selecting a number at least 7. Then find the probability of not selecting a number at least 7.” They must show an outcome list or grid and one complement equation. Collect responses to identify misconceptions for the next lesson.

Resources

• the complete probability lesson deck
• the dyslexia-friendly events and outcomes worksheet
• Six-sided dice or numbered cards
• One simple classroom spinner and board/projector
• Mini-whiteboards, pens, and erasers
• Highlighters for identifying key words
• Coloured pencils for shading outcome grids
• Optional physical or digital random-number generator

Assessment

• Questioning and mini-whiteboard checks assess interpretation of “at least”, “at most”, and inclusive ranges.
• During paired work, check outcome lists, correct sample spaces, simplified probabilities, and explanations of direct versus complementary methods.
• Use the exit response to identify whether students can calculate a complement and communicate the reasoning accurately.

Differentiation

• Provide outcome grids, partially completed sample spaces, highlighted key words, and sentence starters such as “The event includes…” and “The complement is…”.
• Offer physical die or spinner demonstrations before moving to symbolic notation; allow students to rehearse explanations orally with a partner.
• Provide a dyslexia-friendly worksheet version: clear sans-serif font, generous spacing, high contrast, one problem per line, minimal visual clutter, and no unnecessary copying.
• Support EAL learners with a small probability word bank and examples for “at least”, “at most”, “inclusive”, “multiple”, and “complement”. Extension learners justify both direct and complementary solutions to multi-condition LOTTO problems and discuss overlaps and efficiency.

Extension

• Generalise the complement rule by explaining why (P(A)+P(\text{not }A)=1) for every complete sample space.
• Create a new LOTTO event for a partner that can be solved directly and by using a complement, then explain which method is preferable.