Cliste School
Get started

Probability

Venn Diagrams and Set Notation

Year 10 · Year 11

  • ✓Understand and use standard set notation, including ξ, A', A ∪ B, A ∩ B, ∅, ∈.
  • ✓Represent events and their probabilities using Venn diagrams.
  • ✓Calculate probabilities of events and combined events from Venn diagrams.
  • ✓Understand and calculate conditional probabilities using Venn diagrams and the formula P(A|B) = P(A ∩ B) / P(B).
  • ✓Solve problems involving Venn diagrams and conditional probability in context.

Key concepts

Set

A collection of distinct objects or elements.

Universal Set (ξ)

The set of all possible elements or outcomes in a particular context. Represented by a rectangle in a Venn diagram.

Element (∈)

An object or member of a set. For example, if A = {1, 2, 3}, then 1 ∈ A.

Empty Set (∅ or {})

A set containing no elements.

Subset (⊆)

Set A is a subset of set B if every element of A is also an element of B.

Union (A ∪ B)

The set of all elements that are in set A OR in set B (or both). In terms of probability, P(A ∪ B) is the probability that event A occurs OR event B occurs (or both).

Intersection (A ∩ B)

The set of all elements that are in set A AND in set B. In terms of probability, P(A ∩ B) is the probability that event A occurs AND event B occurs.

Complement (A' or Aᶜ)

The set of all elements in the universal set (ξ) that are NOT in set A. In terms of probability, P(A') is the probability that event A does NOT occur.

Venn Diagram

A diagram using overlapping circles to visually represent sets and their relationships within a universal set.

Conditional Probability (P(A|B))

The probability that event A occurs, GIVEN that event B has already occurred. This concept is for the Higher tier.

P(A|B) = P(A ∩ B) / P(B)

Key facts to remember

  • 1ξ represents the universal set.
  • 2A' (or Aᶜ) represents the complement of set A (not A).
  • 3A ∪ B represents the union of sets A and B (A or B or both).
  • 4A ∩ B represents the intersection of sets A and B (A and B).
  • 5∅ (or {}) represents the empty set.
  • 6P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
  • 7P(A') = 1 - P(A).
  • 8P(A|B) = P(A ∩ B) / P(B) (Conditional Probability).

Worked examples

Example 1

In a class of 30 students, 18 study Maths (M), 15 study Physics (P), and 7 study both. a) Draw a Venn diagram to represent this information. b) Find the probability that a randomly chosen student studies Maths only. c) Find the probability that a randomly chosen student studies neither Maths nor Physics.

IDraw a rectangle for the universal set (ξ) and two overlapping circles for M and P.
IIFill in the intersection first: M ∩ P = 7 students.
IIIStudents studying Maths only: 18 - 7 = 11. Place 11 in the M circle, outside the intersection.
IVStudents studying Physics only: 15 - 7 = 8. Place 8 in the P circle, outside the intersection.
VTotal students accounted for: 11 + 7 + 8 = 26.
VIStudents studying neither: 30 - 26 = 4. Place 4 outside both circles but inside the rectangle.
VIIFor part b), P(Maths only) = (Number of students studying Maths only) / (Total students) = 11/30.
VIIIFor part c), P(Neither Maths nor Physics) = (Number of students studying neither) / (Total students) = 4/30 = 2/15.

Answer

a) [Diagram description: Rectangle for ξ. Circle M contains 11 (M only) and 7 (M and P). Circle P contains 8 (P only) and 7 (M and P). 4 is outside both circles.] b) 11/30 c) 2/15

Always start by filling in the intersection of the sets.

Example 2

A survey of 100 people found that 60 like tea (T), 40 like coffee (C), and 25 like both. a) Draw a Venn diagram showing the number of people in each region. b) Find P(T ∪ C). c) Find P(T' ∩ C).

IDraw a rectangle for ξ and two overlapping circles for T and C.
IIFill in the intersection: T ∩ C = 25 people.
IIIPeople who like Tea only: 60 - 25 = 35.
IVPeople who like Coffee only: 40 - 25 = 15.
VTotal accounted for: 35 + 25 + 15 = 75.
VIPeople who like neither: 100 - 75 = 25.
VIIFor part b), P(T ∪ C) = P(Tea only) + P(Coffee only) + P(Both) = (35 + 15 + 25) / 100 = 75/100 = 3/4.
VIIIFor part c), P(T' ∩ C) means people who do NOT like Tea AND DO like Coffee. This is the region for Coffee only. So, P(T' ∩ C) = 15/100 = 3/20.

Answer

a) [Diagram description: Rectangle for ξ. Circle T contains 35 (T only) and 25 (T and C). Circle C contains 15 (C only) and 25 (T and C). 25 is outside both circles.] b) 3/4 c) 3/20

Ensure all regions sum to the total number of elements in the universal set.

Example 3

Using the information from Example 2 (100 people, 60 like tea, 40 like coffee, 25 like both). a) Find the probability that a person likes tea, given that they like coffee. b) Find the probability that a person likes coffee, given that they do not like tea.

IRecall the Venn diagram from Example 2: Tea only = 35, Coffee only = 15, Both = 25, Neither = 25. Total = 100.
IIFor part a), we need P(T|C). Using the formula P(T|C) = P(T ∩ C) / P(C).
IIIP(T ∩ C) = 25/100.
IVP(C) = (People who like Coffee) / (Total people) = (25 + 15) / 100 = 40/100.
VP(T|C) = (25/100) / (40/100) = 25/40 = 5/8.
VIFor part b), we need P(C|T'). Using the formula P(C|T') = P(C ∩ T') / P(T').
VIIP(C ∩ T') is the probability of liking Coffee AND NOT liking Tea, which is the 'Coffee only' region: 15/100.
VIIIP(T') is the probability of NOT liking Tea. This includes people who like Coffee only (15) and people who like neither (25). So, P(T') = (15 + 25) / 100 = 40/100.
9P(C|T') = (15/100) / (40/100) = 15/40 = 3/8.

Answer

a) 5/8 b) 3/8

Conditional probability effectively reduces the sample space to the 'given' event.

Common mistakes

  • ✗Confusing the symbols for union (∪) and intersection (∩).
  • ✗Forgetting to subtract the intersection when calculating the number of elements in 'A only' or 'B only'.
  • ✗Not including elements that are outside all specified sets but still within the universal set.
  • ✗Incorrectly identifying the sample space for conditional probability calculations (e.g., using the total universal set instead of the 'given' event).
  • ✗Misinterpreting 'A or B' as only A or only B, rather than A or B or both.

Exam tips

  • ★Always draw a clear Venn diagram first, even if not explicitly asked. This helps visualise the problem.
  • ★When filling in a Venn diagram, always start with the innermost intersection (e.g., A ∩ B) and work outwards.
  • ★Carefully read the question to identify keywords like 'and' (intersection), 'or' (union), 'not' (complement), and 'given that' (conditional probability).
  • ★Check that the sum of the probabilities (or number of elements) in all regions of your Venn diagram equals 1 (or the total number of elements in the universal set).

Ready to practise?

Try a problem on this topic

Snap a photo or type a question — get step-by-step working instantly.

Solve a problem →← Maths curriculum