3.2.2 Further Venn Diagrams

Venn diagrams are a visual tool used to represent the relationships between different sets, particularly in probability and set theory. They help in visualizing events, their intersections, unions, and complements. Understanding how to interpret and use Venn diagrams is crucial for solving problems involving combined events and probabilities.

Basic Concepts

Set: A collection of distinct objects or elements.
Universal Set ( $\xi$ ): The set that contains all possible outcomes or elements under consideration.
Event: A subset of the universal set, representing specific outcomes.
Intersection ( $A \cap B$ ): The set of elements that are common to both events $A$ and $B$ .
Union ( $A \cup B$ ): The set of elements that are in either event $A$ or event $B$ (or both).
Complement ( $A'$ ): The set of elements that are not in event $A$ .

Venn Diagram Notation

Two-Set Venn Diagram

Represents two events $A$ and $B$ within the universal set $\xi$ .
The overlapping region represents the intersection $A \cap B$ .
The total area of the circles represents the union $A \cup B$ .

Three-Set Venn Diagram

Represents three events $A$ , $B$ , and $C$ .
The diagram includes regions for all possible intersections between these events, including $A \cap B \cap C$ .

Probability Using Venn Diagrams

Venn diagrams are useful for solving problems involving probabilities of combined events.

Key Formulas

Union of Two Events

P(A \cup B) = P(A) + P(B) - P(A \cap B)

This formula accounts for the fact that the intersection $A \cap B$ is counted twice when summing $P(A)$ and $P(B$ ) .

Union of Three Events

P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C)

Complement of an Event

P(A') = 1 - P(A)

Example Problems with Venn Diagrams

infoNote

Example 1: Two-Set Venn Diagram In a class of $40$ students, $25$ study Mathematics ( $M$ ) and $20$ study Physics ( $P$ ).

If $10$ students study both subjects,

Question : What is the probability that a randomly selected student studies either Mathematics or Physics?

Worked Solution

Step 1: Work out $P(M)$ , $P(P)$ and $P(M \cap P)$

P(M) = \frac{25}{40} = 0.625

P(P) = \frac{20}{40} = 0.5

P(M \cap P) = \frac{10}{40} = 0.25

Step 2: Use the Union Formula and conclude

P(M \cup P) = P(M) + P(P) - P(M \cap P)

= 0.625 + 0.5 - 0.25 = 0.875

The probability that a randomly selected student studies either Mathematics or Physics is 0.875.

Venn Diagram Representation

Draw two overlapping circles labeled $M$ and $P$ within a rectangle representing the universal set of 40 students.
Label the intersection (overlap) with 10 students, the only-Math part with 15 students, and the only-Physics part with 10 students. The region outside both circles represents the remaining 5 students who study neither subject.

infoNote

Example 2: Three-Set Venn Diagram Problem: In a survey of 100 people:

40 like Tea ( T ),
50 like Coffee ( C ),
30 like Juice ( J ),
15 like both Tea and Coffee,
10 like both Tea and Juice,
8 like both Coffee and Juice,
5 like all three. Question : Find the number of people who like exactly one of these drinks.

Worked Solution

Step 1: Assign the Given Values:

$|T \cap C \cap J| = 5$
$|T \cap C| = 15$ , so $|T \cap C \cap J'| = 15 - 5 = 10$
$|T \cap J| = 10$ , so $|T \cap J \cap C'| = 10 - 5 = 5$
$|C \cap J| = 8$ , so $|C \cap J \cap T'| = 8 - 5 = 3$

Step 2: Calculate the Number of People in Each Exclusive Category:

|T \cap C' \cap J'| = |T| - (|T \cap C \cap J'| + |T \cap J \cap C'| + |T \cap C \cap J|)

|T \cap C' \cap J'| = 40 - (10 + 5 + 5) = 20

|C \cap T' \cap J'| = |C| - (|T \cap C \cap J'| + |C \cap J \cap T'| + |T \cap C \cap J|)

|C \cap T' \cap J'| = 50 - (10 + 3 + 5) = 32

|J \cap T' \cap C'| = |J| - (|T \cap J \cap C'| + |C \cap J \cap T'| + |T \cap C \cap J|)

|J \cap T' \cap C'| = 30 - (5 + 3 + 5) = 17

Step 3: Summarize the Results:

People who like only Tea: 20
People who like only Coffee: 32
People who like only Juice: 17 Total number of people who like exactly one drink:

20 + 32 + 17 = 69

Venn Diagram Representation:

Draw three overlapping circles labeled $T$ , $C$ , and $J$ .
Fill in the intersection regions with the calculated values and the exclusive regions with the numbers from Step 2.

Applications of Venn Diagrams

Probability: Venn diagrams help in visualizing and calculating the probabilities of combined events, including intersections, unions, and complements.
Set Theory: They are used to solve problems involving unions, intersections, and differences between sets.
Logic and Decision-Making: Venn diagrams help in visualizing logical relationships, especially in decision-making processes where multiple criteria are involved.

Summary

Venn diagrams are a powerful visual tool in statistics and set theory. They simplify the process of understanding and calculating probabilities of combined events. By representing events as sets within a universal set, Venn diagrams make it easier to solve complex problems involving unions, intersections, and complements. The examples provided demonstrate how to apply Venn diagrams to both two-set and three-set problems, showing their practical utility in various statistical analyses.

Further Venn Diagrams Simplified Revision Notes for A-Level AQA Maths Statistics