124i - Venn Diagrams for Combined Events

Lesson 124 — Explicit Instruction: Venn Diagrams for Combined Events

Strand: Probability | Descriptor: AC9M8P02 | Duration: 45 minutes

Learning Intentions

• To construct Venn diagrams to represent two events and all their combinations.
• To use set notation (, , complement) and Venn diagrams to calculate probabilities.

Success Criteria

I can:

1. Construct a two-circle Venn diagram from given information about two events.
2. Use correct set notation: (union), (intersection), (complement).
3. Calculate probabilities of union, intersection and complement events using a Venn diagram.
4. Apply the addition rule .

Warmup

(5 minutes — mini whiteboards, rapid-fire)

1. What does mean, in words?
2. What does mean, in words?
3. What does mean?
4. If and , and and can never happen together, find .

Answers: 1. Both and happen (“intersection”); 2. happens, or happens, or both (“union”); 3. does not happen (the complement of ); 4. — simple addition works only because there is no overlap to double-count, which is the hook for today.

Activities

Activity 1 — Explicit Instruction: Building a Venn Diagram (10 min)

I do: In a class of , like maths, like science, and like both.

Draw two overlapping circles labelled and . Fill the intersection first: . Then work outward:

Read probabilities directly: ; ; .

Introduce the addition rule:

Check against the diagram: . ✓ “We subtract the intersection because adding counts the overlap twice.”

We do: Together build a Venn diagram for people: own a car, own a motorbike, own both. Find , , and two ways (from the diagram, and from the formula).

You do: students: play basketball, play tennis, play both. Build the Venn diagram. Find , , .

(We do answers: car only , neither , . You do answers: basketball only , tennis only , both , neither ; ; ; .)

Activity 2 — Explicit Instruction: Working Algebraically with the Addition Rule (10 min)

I do: , , . Find .

Also: ; .

We do: , , . Rearrange to find .

You do: , , . Find and .

(Answers: ; .)

Activity 3 — Applied Task: Making a Decision from a Venn Diagram (16 min)

Pairs, then whole-class share.

A gym has members. attend yoga classes, attend spin classes, and attend both.

(a) Draw a Venn diagram and label all four regions. (b) Find . (c) Find . (d) Of the members who attend spin, what fraction also attend yoga? (e) Harder. The gym is considering closing the spin class. Using the Venn diagram, find how many members would be lost entirely (i.e. attend spin, but not yoga). What percentage of the total members is this? Should this influence the decision? Justify using the data.

Socratic scaffolding for part (e):

Prompt	Purpose
Understand: what counts as “lost entirely”?	Only members in the spin-only region — members who do both would still have yoga to attend.
Devise a plan	Read the spin-only region from the Venn diagram.
Carry out	Spin only .
Express as a percentage	.
Looking back — is this significant?	One in five members is a substantial minority — a real cost to closing the class — though a full decision would also weigh running costs and revenue, not probability alone.

Answers: Regions: yoga only , spin only , both , neither . (b) . (c) . (d) . (e) members, — a data-grounded reason for caution before closing the class.

Checks for Understanding

(6 minutes — exit ticket, collected)

students: like reading, like gaming, like both.

1. Find the four Venn diagram regions: reading only, gaming only, both, neither.
2. Find .
3. Find .
4. Write, using set notation, an expression for “likes gaming but not reading”, and evaluate it.

Answers: 1. Reading only , gaming only , both , neither ; 2. ; 3. ; 4. , evaluated as .

Common Misconceptions

Misconception	How to pre-empt it
Adding without subtracting the overlap.	Always ask “does this double-count anyone?” before adding. Show the doubled intersection visually.
Writing the given overlap total directly into the “only” regions.	Insist the rule “fill the intersection first, then subtract to find each ‘only’ region” every time a diagram is built.
Confusing (intersection, “and”) with (union, “or”).	Link to the pointed “cap” shape of the overlap region; to the wide cup shape of everything covered.
Assuming “neither” is automatically zero.	Always calculate neither explicitly; it is often nonzero.
Treating and as the same value.	Contrast (the whole circle) with (circle minus overlap) side by side.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). In a survey, people like tea only, like coffee only, like both, and like neither. If the total surveyed is , , , and , find .

Answer

E2 (Kangaroo style). and . Find the smallest and largest possible values of .

Answer

Largest: (the overlap cannot exceed either circle).

Smallest: since , .

So .

E3 (Challenge). , , . Find and .

Answer

Homework

1. people were surveyed: like coffee, like tea, like both. Find all four Venn diagram regions and .
2. Using Q1, find and .
3. , , . Find .
4. , , . Find .
5. Reasoning. Explain why only when and cannot happen together (i.e. ). Give an example of two such events.
6. Challenge. In a group of students, every student studies at least one of French or Spanish. study French and study Spanish. How many study both? What percentage of the group is this?

Answers: 1. Both , coffee only , tea only , neither ; . 2. ; . 3. . 4. . 5. If and can never both occur, there is no shared region to double-count, so simple addition already gives the correct total — e.g. rolling a die, and cannot occur on the same roll. 6. Since nobody studies neither, , so , giving both , which is of the group.