Practice Maths

Events, Sets and Venn Diagrams

Key Terms

The sample space (ξ or S) is the set of all possible outcomes of an experiment.
An event is a subset of the sample space — a collection of outcomes we are interested in.
The complement of event A, written A′ (or A̅ or Ac), contains all outcomes in the sample space not in A.
The union A ∪ B contains all outcomes in A or B (or both).
The intersection A ∩ B contains all outcomes in both A and B.
Events are mutually exclusive if they cannot occur simultaneously: A ∩ B = ∅ (empty set).
A Venn diagram uses overlapping circles inside a rectangle (ξ) to show relationships between events.
In a Venn diagram with counts: P(A) = (count in A) ÷ (total count).
Key Formulas
• P(A′) = 1 − P(A)
• P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• Mutually exclusive: P(A ∪ B) = P(A) + P(B)  [since P(A ∩ B) = 0]
• P(ξ) = 1   P(∅) = 0
• 0 ≤ P(A) ≤ 1 for any event A

Venn diagram showing A, B, A ∩ B, A ∪ B and A′

ξ A B A∩B A∪B (everything in both circles) A′ A′
Hot Tip When filling in a Venn diagram with given information, always start with the intersection (A ∩ B) first, then work outward. If you know n(A) and n(A ∩ B), then the part of A only = n(A) − n(A ∩ B). Finally, the complement = total − everything inside the circles.

Worked Example 1 — Reading a Venn diagram

Question: In a class of 30 students, 18 study Biology (B), 14 study Chemistry (C), and 8 study both. Find P(B′), P(B ∪ C) and P(B ∩ C′).

Step 1 — Fill in the Venn diagram:

• Both: 8  • Biology only: 18 − 8 = 10  • Chemistry only: 14 − 8 = 6  • Neither: 30 − 10 − 8 − 6 = 6

Step 2 — Calculate:

P(B′) = 1 − 18/30 = 12/30 = 2/5

P(B ∪ C) = (10 + 8 + 6)/30 = 24/30 = 4/5

P(B ∩ C′) = Biology only / total = 10/30 = 1/3

Worked Example 2 — Three-set Venn diagram

Question: 50 people were surveyed about apps they use: 28 use Facebook (F), 22 use Instagram (I), 18 use TikTok (T), 10 use F and I, 8 use F and T, 6 use I and T, and 4 use all three. Find P(F ∪ I ∪ T) and the number using none.

Step 1 — Use the inclusion-exclusion principle:

n(F ∪ I ∪ T) = 28 + 22 + 18 − 10 − 8 − 6 + 4 = 48

Step 2: Using none = 50 − 48 = 2

P(F ∪ I ∪ T) = 48/50 = 24/25

Set Notation and Language: Precision Matters

Probability theory uses the language of sets because events are, mathematically, just subsets of the sample space. The three fundamental operations are union, intersection, and complement, and they directly correspond to everyday language:

A ∪ B (“A union B”) = A OR B = at least one of A, B occurs. In everyday language: “the customer orders coffee or tea” means coffee, tea, or both. In probability, “or” is inclusive — both is fine.

A ∩ B (“A intersect B”) = A AND B = both A and B occur. More restrictive: the customer orders both coffee AND tea.

A′ (“A complement”) = NOT A = everything except A. If A is “rolling an even number”, then A′ is “rolling an odd number”.

The sample space ξ contains all possible outcomes: P(ξ) = 1 always. The empty set ∅ contains no outcomes: P(∅) = 0. Every event A satisfies 0 ≤ P(A) ≤ 1.

Venn Diagrams: Visualising Relationships

A two-event Venn diagram has four regions: “A only” (inside A, outside B), “B only” (inside B, outside A), “both A and B” (the intersection, inside both circles), and “neither” (outside both circles, inside the rectangle).

When you are given numerical information about a two-event experiment, always fill in the Venn diagram systematically. Start with the intersection: if you know P(A ∩ B) or n(A ∩ B), put that in the overlap region first. Then “A only” = n(A) − n(A ∩ B), “B only” = n(B) − n(A ∩ B). Finally “neither” = total − everything in the circles. Working inside-out prevents double-counting errors.

Adding Probabilities: Why We Subtract the Intersection

The addition formula P(A ∪ B) = P(A) + P(B) − P(A ∩ B) corrects for double-counting. When we add P(A) and P(B), the intersection P(A ∩ B) has been counted twice: once in P(A) and once in P(B). Subtracting P(A ∩ B) once brings us back to counting it exactly once.

Imagine counting people who like pizza or pasta. If 15 like pizza, 12 like pasta, and 5 like both: adding 15 + 12 = 27 counts the 5 “both” people twice. The correct count is 27 − 5 = 22 people who like at least one. This is the inclusion-exclusion principle in action.

Mutually Exclusive Events: No Overlap Possible

Events A and B are mutually exclusive if they cannot both happen at the same time: A ∩ B = ∅. In a Venn diagram, the two circles do not overlap at all. Rolling an even number and rolling an odd number are mutually exclusive: they cannot both occur on the same roll.

For mutually exclusive events: P(A ∪ B) = P(A) + P(B) (no subtraction needed since P(A ∩ B) = 0). This is the simple addition rule that applies only when events are mutually exclusive. Always check whether events are mutually exclusive before using this simplified formula.

The Complement Rule: Total Probability Is Always 1

Since every outcome is either in A or not in A, and these two groups cover all possible outcomes with no overlap: P(A) + P(A′) = 1, so P(A′) = 1 − P(A). This is extremely useful when “A does not occur” is easier to compute than “A occurs”. For instance, P(at least one success) = 1 − P(no successes) is often much simpler to calculate directly.

Exam Tip: In probability, “or” almost always means the inclusive “or” (A or B or both). Use the union A ∪ B and apply the addition formula P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The only time you use the simple P(A) + P(B) is when the events are mutually exclusive and the intersection is zero. Always check for mutual exclusivity before simplifying.
Exam Tip: The inclusion-exclusion formula extends to three events: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(A ∩ C) − P(B ∩ C) + P(A ∩ B ∩ C). For a three-circle Venn diagram, always start by filling in the three-way intersection (centre region), then the three pairwise-only regions, then the three “one event only” regions, and finally the “none” region.

Mastery Practice

See Answers ➔
  1. Fluency

    Use set notation to describe the shaded region in each Venn diagram. Choose from: A, B, A′, B′, A ∪ B, A ∩ B, A ∩ B′, A′ ∩ B, (A ∪ B)′.

    1. (a) Only the left circle (A) is shaded, excluding the overlap.
    2. (b) The overlap region only is shaded.
    3. (c) Everything outside both circles is shaded.
    4. (d) The entire right circle (B) and the overlap are shaded.
  2. Fluency

    A die is rolled once. The sample space is {1, 2, 3, 4, 5, 6}. Let A = {even numbers} and B = {numbers greater than 3}.

    1. (a) List the elements of A, B, A ∩ B, and A ∪ B.
    2. (b) Find P(A), P(B), P(A ∩ B), and P(A ∪ B).
    3. (c) Find P(A′) and P(B′).
    4. (d) Are A and B mutually exclusive? Explain.
  3. Fluency

    From the Venn diagram below, where the numbers represent counts of students:

    Sport only: 12  |  Both Sport and Music: 5  |  Music only: 9  |  Neither: 4  |  Total: 30

    1. (a) Find P(Sport).
    2. (b) Find P(Music).
    3. (c) Find P(Sport ∩ Music).
    4. (d) Find P(Sport ∪ Music).
    5. (e) Find P(Sport′).
  4. Fluency

    State whether the following pairs of events are mutually exclusive. If not, give an example of an outcome in both.

    1. (a) Rolling a 6 and rolling an odd number on one die.
    2. (b) Drawing a red card and drawing a king from a standard deck.
    3. (c) Selecting a multiple of 3 and a multiple of 5 from {1, 2, ..., 20}.
    4. (d) Being born in January and being born in a month with 31 days.
  5. Understanding

    In a group of 40 people, 25 like coffee (C), 18 like tea (T), and 8 like both.

    Method: Draw a Venn diagram. Fill in the intersection first, then each region individually, then find the “neither” group.
    1. (a) Draw and label a Venn diagram with counts in each region.
    2. (b) Find the probability that a randomly selected person likes coffee but not tea.
    3. (c) Find P(C ∪ T).
    4. (d) Find the probability that a person likes neither coffee nor tea.
    5. (e) Find P(C′ ∩ T).
  6. Understanding

    P(A) = 0.5, P(B) = 0.4, and P(A ∩ B) = 0.2.

    1. (a) Find P(A ∪ B).
    2. (b) Find P(A′).
    3. (c) Find P(A ∩ B′) (i.e., A only).
    4. (d) Find P(A′ ∩ B′) (neither A nor B).
    5. (e) Verify that all four regions of your Venn diagram add to 1.
  7. Understanding

    Three-set Venn diagram.

    Inclusion-exclusion for 3 sets: n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(A∩C) − n(B∩C) + n(A∩B∩C)

    In a survey of 60 students: 30 play Guitar (G), 24 play Piano (P), 20 play Drums (D). 12 play G and P, 8 play G and D, 6 play P and D, and 4 play all three.

    1. (a) Find n(G ∪ P ∪ D).
    2. (b) How many students play none of the three instruments?
    3. (c) Find P(G only).
    4. (d) Find P(exactly one instrument).
  8. Understanding

    Working backwards from probabilities.

    P(A) = 0.6, P(A ∪ B) = 0.85, and A and B are not mutually exclusive.

    1. (a) Find P(A ∩ B) if P(B) = 0.45.
    2. (b) Find P(A′ ∩ B) (B only).
    3. (c) What is the probability of neither A nor B?
  9. Problem Solving

    Health survey.

    Challenge. A clinic surveyed 200 patients for three conditions: Hypertension (H), Diabetes (D), and Obesity (O). Results: n(H)=90, n(D)=70, n(O)=60, n(H∩D)=30, n(H∩O)=25, n(D∩O)=20, n(H∩D∩O)=10.
    1. (a) Find the number of patients with at least one condition.
    2. (b) Find the number with none of the three conditions.
    3. (c) Find the probability that a patient has exactly one condition.
    4. (d) Find P(H only).
  10. Problem Solving

    Logical deduction with set notation.

    Challenge.
    1. (a) If P(A) = 0.4 and A and B are mutually exclusive with P(A ∪ B) = 0.7, find P(B).
    2. (b) If P(A) = p, P(B) = 2p, and P(A ∩ B) = p/3, and P(A ∪ B) = 0.8, find p.
    3. (c) Events A, B, C are mutually exclusive and exhaustive with P(A) = 2P(B) and P(C) = 3P(B). Find P(A), P(B), P(C).