Complementary Events

Overview

In this lesson, students learn what complementary events are and how they connect to sample spaces and probability. They discuss examples, then model complementary outcomes using a simple experiment and class reasoning.

Learning intentions

Students will be able to:

• identify complementary events in a single-stage situation
• explain how complementary events relate to “all outcomes” in the sample space
• assign probabilities to complementary events and use this to make predictions about likely outcomes

Success criteria

Students can:

• give at least one example of two complementary events
• state the sample space for a single-stage event (e.g. coin, dice, spinner)
• explain why complementary event probabilities add to 1
• use relative frequency from a short trial to check their expectations (at a basic level)

Curriculum links

• Probability — identify sample spaces for single-stage events; assign probabilities; predict relative frequencies - (Builds understanding used in) repeated chance experiments and simulations by comparing predictions with observed results - (Optional link for later lessons) analysing results using displays and summary measures (supports ST strand work)

Lesson structure (60 minutes)

1. 0–5 min · Conversation hook Teacher asks: “If one event happens, what must be true about the other event?” Students think-pair-share and try to say examples using everyday language (e.g. “heads” vs “not heads”).

2. 5–15 min · Direct teach: complementary events Teacher introduces complementary events: two events that cannot both happen in the same trial and together include all outcomes in the sample space. Students copy a short definition and example table in their workbook. Key example (coin): Sample space {Heads, Tails}.

• Event A: “Heads”
• Event B: “Tails” Teacher emphasises: A and B are complementary; P(A) + P(B) = 1.

3. 15–25 min · Guided practice: sample spaces first Teacher displays three situations (students work in pairs, then discuss as a class): a) Dice roll, event A: “roll a 2”, event B: “not a 2” b) Spinner with 4 equal sections: event A: “Land on red”, event B: “Land on not red” c) Lucky dip with 5 slips: 2 are green, 3 are blue; event A: “green”, event B: “not green” Students write the sample space, name complementary events, then state why they are complementary in one sentence.

4. 25–35 min · Probability check with predictions Teacher models using numbers:

• For dice, P(2) = 1/6, so P(not 2) = 5/6 Teacher asks students to verify complementary addition: “What should the sum be?” Students calculate answers and hold up a finger for “sum is 1” (quick visual check).

5. 35–48 min · Short experiment to connect theory and relative frequency Teacher sets up a quick class trial (single-stage repeated trials using a coin or a die rolling station). Choose one:

• Option A: 30 coin flips recorded by groups
• Option B: 24 die rolls recorded by groups Students tally outcomes for one event (e.g. Heads or “rolling 6”) and then compute the relative frequency for that event and its complement. Teacher circulates to ensure students compare predicted probability with observed relative frequency (focus on closeness, not perfect results).

6. 48–57 min · Whole-class wrap: explain differences Teacher leads a discussion: “Why might our relative frequency not exactly match probability after only a small number of trials?” Students respond using sentence frames (provided):

• “Our result was different because…”
• “As we do more trials, we would expect…” Teacher reinforces the idea that randomness causes variation.

7. 57–60 min · Exit ticket (quick check) Students answer one prompt on a slip: “A die is rolled once. Event A: roll an even number. Event B: roll an odd number. (a) Are these complementary? Explain briefly. (b) Find P(A) and P(B).”

Resources

• coin (or digital coin tool) and/or dice
• spinner template with clearly equal sections (optional)
• dice/spinner/lucky dip scenario cards
• recording sheets for tallies and relative frequency (simple tables)
• whiteboard, markers, or interactive display
• sentence starter strips for explanations
• dyslexia-friendly printed task sheets (large spacing, fewer lines)

Assessment

• Formative: teacher listens during pair discussions for correct complementary definitions and correct sample space reasoning
• Formative: checks calculations of probabilities and whether students state the sum equals 1
• Exit ticket: determines mastery of complementary identification and basic probability assignment

Differentiation

• Visual aids: use colour-coding on sample space diagrams (e.g. highlight the event, shade the complement) to support concept clarity
• Scaffolded language: provide sentence frames for “Complementary events are…” and “They are complementary because…”
• Dyslexia-friendly options: provide a simplified one-page handout version with:
• short definitions
• bold key terms (probability, sample space, event, complementary)
• increased line spacing and minimal text per box
• option to respond orally or with a checklist for the exit ticket section (teacher transcribes if needed)
• Support for wide ability:
• Support: only require Event and Complement identification plus probability for one event (e.g. P(event) from a simple fraction)
• Extension for fast finishers (still within lesson goals): ask students to create their own complementary-event pair from a given context and justify using sample space