Venn Diagrams

Video coming soon3 worked examples

Theory

A Venn diagram is a powerful visual tool used to organise data and show how different items fit into specific categories (or sets). They make calculating complex probabilities much easier.

1. The Structure of a Venn Diagram

The Sample Space: The large outer rectangle represents the entire sample space (every single person or item surveyed).
The Sets: The circles inside represent the specific categories.
The Complement: Any number written outside the circles but inside the rectangle represents items that do not belong to any of the categories (the "neither" group).

2. Key Terminology and Notation

You must be familiar with the formal set notation used to describe different parts of the diagram:

Intersection ( $A \cap B$ ): The overlapping section in the middle. This represents items that are in Both A AND B.
Union ( $A \cup B$ ): The entire area covered by the circles combined. This represents items that are in A OR B (or both).
Complement ( $A'$ ): This means NOT A. It includes everything in the diagram that is outside of circle A.

3. Constructing Venn Diagrams (The Subtraction Trap)

Massive Exam Trap:

When you are given the total number of people in a category, that total includes the people in the overlapping sections. You must always subtract the intersection from the group totals to find the number of people who belong to "only" one category.

Strategy: Whenever you construct a Venn diagram, always start from the very centre (the intersection of all circles) and work your way outwards, subtracting as you go.

Worked examples

Example 1

Example 1: Interpreting a Venn Diagram & Notation

A gym surveyed 80 of its members to see if they regularly attend the Spin class (S) or the Yoga class (Y). The results are shown in the Venn diagram below:

Spin circle only: 25
Yoga circle only: 20
Intersection (middle): 15
Outside the circles: 20

Calculate the following probabilities:

(a) $P(S \cap Y)$

(b) $P(Y')$

(a) $P(S \cap Y)$ means the probability of Spin AND Yoga (the intersection).

$P(S \cap Y) = \frac{15}{80} = \frac{3}{16}$ (or 0.1875).

(b) $P(Y')$ means the probability of NOT Yoga. We must add everything outside the Yoga circle (Spin only + Neither).

$P(Y') = \frac{25 + 20}{80} = \frac{45}{80} = \frac{9}{16}$ (or 0.5625).

Probability = $\frac{25 + 20}{80} = \frac{45}{80} = \frac{9}{16}$ .

Example 2

Example 2: Constructing a 2-Circle Venn Diagram

A veterinary clinic reviews the records of 120 dogs.

70 dogs are vaccinated against rabies.
55 dogs are microchipped.
20 dogs are neither vaccinated nor microchipped.

(a) Construct a Venn diagram to represent this information.

(b) Calculate the probability that a randomly selected dog is microchipped, but NOT vaccinated.

(a)

Find the intersection first: If 20 are neither, then $120 - 20 = 100$ dogs have at least one treatment. However, our totals add up to $70 + 55 = 125$ . The overlap is $125 - 100 = 25$ dogs. (This goes in the middle).
Find the "only" sections: Vaccinated ONLY = $70 - 25 = 45$ . Microchipped ONLY = $55 - 25 = 30$ .
Draw the diagram: Draw a rectangle containing two overlapping circles (V and M). Put 25 in the middle, 45 in the V-only section, 30 in the M-only section, and 20 outside the circles.

(b) Microchipped but NOT vaccinated is the "Microchipped ONLY" section.

Probability = $\frac{30}{120}$ (or $\frac{1}{4}$ or 0.25).

Example 3

Example 3: Constructing a 3-Circle Venn Diagram

A group of 100 university students were asked which streaming services they subscribe to: Netflix (N), Amazon Prime (A), and Disney+ (D).

10 subscribe to all three.
25 subscribe to Netflix and Amazon.
20 subscribe to Amazon and Disney+.
15 subscribe to Netflix and Disney+.
50 subscribe to Netflix in total.
45 subscribe to Amazon in total.
40 subscribe to Disney+ in total.

(a) Construct a Venn diagram to show this information.

(b) Calculate the probability that a student chosen at random subscribes to none of these services.

(a) Strategy: Start at the centre and work outwards!

All three: 10 goes in the very centre.
Intersections of two: N & A only: $25 - 10 = 15$ . A & D only: $20 - 10 = 10$ . N & D only: $15 - 10 = 5$ .
Individual circles (subtracting everything already in that circle): N only: $50 - (15 + 10 + 5) = 20$ . A only: $45 - (15 + 10 + 10) = 10$ . D only: $40 - (10 + 10 + 5) = 15$ .

(Your diagram should have three overlapping circles with these bold numbers placed in the respective sections).

(b) First, find how many students are inside the circles by adding all the sections together:

$10 + 15 + 10 + 5 + 20 + 10 + 15 = 85$ students.

This means $100 - 85 = 15$ students subscribe to none of the services (this number goes outside the circles).

Therefore, the probability is $\frac{15}{100}$ (or $\frac{3}{20}$ or 0.15).

Ready to practise?

10 Venn Diagrams questions with answers and video solutions