119i - Complementary Events and the Sum of Probabilities

Lesson 119 — Complementary Events and the Sum of Probabilities

Strand: Probability | Descriptor: AC9M8P01 | Duration: 45 minutes

Block note. Lessons 119–120 build the foundation for Lesson 121’s consolidation. Today establishes that two complementary events have probabilities summing to one; Lesson 120 extends this into slightly more complex applied contexts, including an introduction to “at least one” problems, before Lesson 121 consolidates with full problem-solving.

Learning Intentions

• To understand that two events are complementary when, together, they account for every possible outcome with no overlap.
• To use to calculate the probability that an event does not occur.

Success Criteria

I can:

1. Explain what makes two events complementary.
2. State the complement of a given event, in words and using symbols.
3. Calculate given , as a fraction, decimal or percentage.
4. Identify when two “opposite-looking” events are not true complements.

Warmup

(5 minutes — complements or not? pairs)

Decide whether each pair of events is truly complementary, and justify your answer.

1. Rolling a die: “even” and “odd”.
2. Rolling a die: “greater than ” and “less than “.
3. Tossing a coin: “heads” and “tails”.
4. Tomorrow’s weather: “raining” and “sunny”.
5. Drawing a card from a standard deck: “red” and “black”.

Answers: 1. Complements — and together cover all outcomes with no overlap; 2. Not complements — together they only cover , missing and ; 3. Complements; 4. Not necessarily complements — a day could be neither (e.g. overcast, without rain); 5. Complements — every card in a standard deck is either red or black.

The pattern that matters today: two events are true complements only if, together, they cover every possible outcome, with no overlap.

Activities

Activity 1 — Explicit Instruction: Defining Complementary Events (12 min)

I do / We do / You do.

Definition. For an event , its complement, written (or “not ”), consists of every outcome not in . Because and together account for the entire sample space with no overlap:

I do: A fair die is rolled. Event = “rolling a “.

Check: ✓

We do: A weather forecast gives . Find .

You do:

1. A spinner has . Find .
2. A sports team’s outcomes are only win or lose (no draw). . Find .
3. A bag of counters has . Find .

(Answers: 1. ; 2. ; 3. .)

Activity 2 — Guided Practice: Complements across Dice, Weather and Sport (16 min)

Pairs. Every answer needs working shown.

1. A weather app states . Find .
2. A fair die is rolled. Find .
3. A football match can end in a win, draw or loss for the home team. , . Find . Then find , and explain why this is not the same as alone.
4. A spinner has sectors coloured red (), blue () and green () — sectors in total. Find two ways: (a) by adding ; (b) using the complement rule . Confirm the two methods agree.
5. In a raffle, . Find , as a percentage correct to decimal place.

Answers:

The key teaching point for Q3: “not a win” includes every non-winning outcome — draw and loss — so it is never the same as “lose” alone whenever a third outcome (like a draw) is possible.

Activity 3 — Inquiry: Constructing True Complements (8 min)

Pairs.

For each event, write its complement precisely in words, express both probabilities symbolically, and check they sum to .

1. Rolling a number greater than on a die.
2. Drawing a card that is not a heart. (What is the complement of this event?)
3. Getting at least heads in coin tosses.
4. Arriving at school before am.

Answers: 1. Complement: “rolling or less” — not “less than ”, a common boundary error. 2. This event is itself the complement of “drawing a heart”; its own complement is “drawing a heart”. 3. Complement: “at most heads” (i.e. , or heads) — not “no heads”. 4. Complement: “arriving at am or later”.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. If , find .
2. State, in words, the complement of “rolling less than on a die”.
3. A game has outcomes win, draw, loss with , . Find and . Explain why they differ.
4. A bag has red, blue and yellow counters only. , . Find , and hence .
5. Reasoning. Explain why “raining” and “sunny” are not true complements of each other, giving an example of a possible outcome that is neither.

Answers: 1. ; 2. “Rolling or more” (i.e. or ); 3. ; (= draw + loss); they differ because “not win” includes both the draw and the loss outcomes, not loss alone; 4. ; ; 5. A day could be overcast — neither raining nor sunny — so the two events do not together cover every possible outcome, meaning in general.

Common Misconceptions

Misconception	How to pre-empt it
Treating the complement of “win” as “lose”, ignoring a possible draw.	Activity 2 Q3’s direct contrast between and .
Writing the complement of “greater than ” as “less than ” instead of ” or less”.	Explicit boundary check in Activity 3 Q1 — always test whether the boundary value itself belongs to or .
Assuming any two “opposite-sounding” events are automatically complements.	Warmup’s rain/sunny and / counterexamples, checked for full coverage and no overlap.
Not checking that stated probabilities actually sum to .	Build the check ”?” into every worked example as a habitual final step.
Believing only applies to coins and dice.	Use varied real contexts (weather, sport, raffles) throughout to show the rule is general.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). . Write as a fraction in simplest form, and find as a fraction.

Answer

, so .

E2 (AMC Junior style). A game has three outcomes: win, draw, loss. and . Find .

Answer

E3 (Challenge). A spinner has equally likely sectors, some red. If and there are red sectors, find .

Answer

. Since sectors represent of the total, sectors.

E4 (Challenge). A probability table lists four outcomes with probabilities , , and . Find , and then find the probability of “not the third outcome”.

Answer

Homework

1. . Find .
2. A die is rolled. Find .
3. A weather forecast gives for a day with only two possible states, sunny or not sunny. Find .
4. A netball team’s results are win, draw or loss. , . Find and .
5. A bag has red, blue, green and yellow counters. , , . Find .
6. State the complement, in words, of “scoring at least on a test”.
7. Reasoning. Explain why “not rolling an even number” and “rolling an odd number” describe the same event on a standard die, but this would not be true if the die had a blank (unnumbered) face.
8. Reasoning. A student says: “If , then .” Explain what assumption this requires, and describe a realistic situation where it would be false.
9. Challenge. Three mutually exclusive and exhaustive outcomes have probabilities in the ratio . Find each probability, and verify they sum to .

Answers: Q1 — . Q2 — , so . Q3 — . Q4 — ; ( draw loss). Q5 — . Q6 — “scoring below “. Q7 — on a standard -face die, every face is numbered, so “not even” and “odd” describe exactly the same set of outcomes ; with a blank face, “not even” would also include the blank outcome, making it a larger event than “odd” alone. Q8 — it assumes the match can only end in a win for A or a win for B, with no possibility of a draw; this would be false in any sport where a draw is possible (e.g. soccer, or netball ending level), since then would be less than . Q9 — total parts ; probabilities are , , ; sum ✓.