What is a Venn Diagram and how is it used in IB MYP Mathematics?

A Venn Diagram is a visual tool used in Mathematics to illustrate the relationships between different sets. In the IB MYP curriculum, it helps students understand concepts in Statistics and Probability by showing how sets intersect, overlap, or remain distinct. This can be particularly useful for solving problems related to union, intersection, and complement of sets.

How do you interpret the intersection of sets in a Venn Diagram?

In a Venn Diagram, the intersection of sets is represented by the overlapping area between circles. This area contains elements that are common to all the sets involved. For example, if Set A and Set B overlap, the intersection represents elements that are in both Set A and Set B. Understanding this concept is crucial for solving problems in Statistics and Probability within the IB MYP framework.

What are some common applications of Venn Diagrams in Statistics and Probability?

Venn Diagrams are commonly used in Statistics and Probability to solve problems involving set theory, such as finding the union, intersection, and complement of sets. They are also used to visualize data, analyze survey results, and solve probability problems by illustrating possible outcomes and their relationships. These applications are integral to the IB MYP Mathematics curriculum, helping students develop critical thinking and analytical skills.

How can Venn Diagrams help in understanding probability concepts in the IB MYP curriculum?

Venn Diagrams help students visualize and understand probability concepts by clearly showing the relationships between different events. For instance, they can illustrate mutually exclusive events, independent events, and conditional probabilities. By using Venn Diagrams, IB MYP students can better grasp how probabilities are calculated and how different events relate to each other in a probability space.

What are the key differences between a two-set and a three-set Venn Diagram?

A two-set Venn Diagram consists of two overlapping circles, representing two sets and their possible relationships, such as union and intersection. A three-set Venn Diagram includes three circles, allowing for more complex relationships and intersections among three sets. This complexity enables students to explore more advanced concepts in Statistics and Probability, such as analyzing data from multiple sources or understanding more intricate probability scenarios, which are part of the IB MYP Mathematics curriculum.

Why is "Filling 'A only' region with $|A|$ instead of $|A| - |A \cap B|$." a common mistake in Venn Diagrams?

Why it happens: Confusing label with region. How to avoid it: 'A' includes the overlap. 'A only' is the part of A outside B. Subtract the intersection.

Why is "Skipping the 'neither' region." a common mistake in Venn Diagrams?

Why it happens: Focus on the circles. How to avoid it: The 'neither' region is outside the circles. Always include it; the four regions sum to the total.

Statistics and ProbabilityIB MYP Maths Extended Level

Venn Diagrams

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Venn Diagrams (Probability) — IB MYP Mathematics (Extended): visualising events

Venn diagrams help compute probabilities for combined events. This MYP Mathematics Extended note covers two-set and three-set Venn diagrams used in probability.

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Fill in a Venn diagram from given probabilities or counts.
MYP Mathematics A — Compute probabilities of combined events.
MYP Mathematics C — Show clear Venn diagrams with all regions filled.

Two-set Venn diagram

Four regions to fill: A only, B only, both, neither.

A two-set Venn diagram has FOUR regions:

$A \cap B$ : in both.
$A$ only: in $A$ but not $B$ ( $A \setminus B$ ).
$B$ only: in $B$ but not $A$ .
Neither: outside both circles (but inside the universal set rectangle).

These four numbers add to the TOTAL (the universal set size).

Strategy for filling:

Start with $|A \cap B|$ (the intersection — often given).
Subtract from $|A|$ to get ' $A$ only'.
Subtract from $|B|$ to get ' $B$ only'.
Subtract everything from the total to get 'neither'.

Worked example. A class of 30 students: 18 like maths, 14 like art, 8 like both. Find $P(\text{likes only art})$ for a random student.

$A \cap B = 8$ .
Maths only: $18 - 8 = 10$ .
Art only: $14 - 8 = 6$ .
Neither: $30 - (10 + 8 + 6) = 6$ .
$P(\text{art only}) = 6/30 = 1/5$ .

Four regions in a two-set Venn.
Start from the centre (intersection), work outwards.
Sum of regions = total.

Three-set Venn diagram

Eight regions including 'neither'.

A three-set Venn has EIGHT regions (seven inside, one outside).

Strategy:

Start from $A \cap B \cap C$ (the centre).
Fill the pairwise overlaps that aren't in the centre.
Fill the 'only' regions.
Fill 'neither / outside'.

Worked example. Survey of 100 students:

50 play football
40 play basketball
30 play tennis
20 play football AND basketball
15 play football AND tennis
10 play basketball AND tennis
5 play all three.

Find how many play NONE.

Start from centre: $|F \cap B \cap T| = 5$ .

Pairwise (NOT in centre):

F and B only: $20 - 5 = 15$ .
F and T only: $15 - 5 = 10$ .
B and T only: $10 - 5 = 5$ .

Single-set 'only':

F only: $50 - (15 + 10 + 5) = 20$ .
B only: $40 - (15 + 5 + 5) = 15$ .
T only: $30 - (10 + 5 + 5) = 10$ .

Total inside at least one circle: $5 + 15 + 10 + 5 + 20 + 15 + 10 = 80$ .

Outside (none): $100 - 80 = 20$ .

So $P(\text{none}) = 20/100 = 1/5$ .

8 regions in a three-set Venn (7 inside + outside).
Always start from the centre (3-way intersection).
Work inside-out.

Using Venn diagrams for probability

Translate counts into probabilities.

Once the Venn is filled in, probability questions become 'count the regions you want' divided by total.

Common probability questions and the corresponding regions:

$P(A \cap B)$ : intersection (centre overlap).
$P(A \cup B)$ : everything inside either circle. (Equivalent: total $-$ neither.)
$P(A \setminus B)$ : $A$ only.
$P(A')$ : outside $A$ (which is $B$ only + neither).
$P(\text{exactly one}) = P(A \text{ only}) + P(B \text{ only})$ .

Worked example (continuing from above). What is $P(\text{at least two sports})$ ?

'At least two' = football+basketball-only ( $15$ ) + football+tennis-only ( $10$ ) + basketball+tennis-only ( $5$ ) + all three ( $5$ ).
Total: $15 + 10 + 5 + 5 = 35$ .
$P = 35/100 = 0.35$ .

A criterion-D question often gives a SURVEY data set and asks compound probability questions. Always:

Draw the Venn diagram.
Fill in ALL regions.
Read off the required count.
Divide by the total to get the probability.

Fill all regions FIRST.
Read off the count in the required region(s).
Divide by total to get probability.

How it’s examined

Criterion A: fill in Venn diagrams and compute probabilities. Criterion C: clear, labelled Venn diagrams. Criterion D: real survey data with compound probability questions.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

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Step-by-step worked examples — Venn Diagrams

Step-by-step solutions to past-paper-style questions on venn diagrams, written exactly the way a tutor would explain them at the board.

1Two-set Venn
Building confidence• two-set
Question
In a survey of 50 people: 30 like tea, 25 like coffee, 12 like both. (a) How many like tea only? (b) How many like neither?
Step-by-step solution
1. Step 1
  Both: 12. Tea only: $30 - 12 = 18$ .
2. Step 2
  Coffee only: $25 - 12 = 13$ .
3. Step 3
  Inside any circle: $18 + 12 + 13 = 43$ .
4. Step 4
  Neither: $50 - 43 = 7$ .
Answer
(a) 18 like tea only. (b) 7 like neither.
2Three-set Venn
Stretch• three-set
Question
$|A| = 25$ , $|B| = 30$ , $|C| = 20$ , $|A \cap B| = 10$ , $|A \cap C| = 8$ , $|B \cap C| = 12$ , $|A \cap B \cap C| = 5$ . Find the number in $A$ only.
Step-by-step solution
1. Step 1
  From $|A \cap B| = 10$ , the centre is 5, so 'A and B only' (not C) is $10 - 5 = 5$ .
2. Step 2
  From $|A \cap C| = 8$ , 'A and C only' = $8 - 5 = 3$ .
3. Step 3
  From $|A| = 25$ : A only = $25 - (5 + 3 + 5) = 25 - 13 = 12$ .
Answer
12 students are in $A$ only.
3Probability from Venn
Stretch• probability
Question
From the two-set Venn data above (50 people: 18 tea only, 12 both, 13 coffee only, 7 neither), find $P(\text{likes tea | likes coffee})$ .
Step-by-step solution
1. Step 1
  Conditional: restrict to coffee-likers. Total coffee-likers: $12 + 13 = 25$ .
2. Step 2
  Of those, $|T \cap C| = 12$ .
3. Step 3
  $P(\text{tea | coffee}) = 12/25$ .
Answer
$12/25 = 0.48$

Key Definitions and Keywords — Venn Diagrams

Definitions to memorise and the exact keywords mark schemes credit for venn diagrams answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Intersection (probability)
Examiner keyword
$P(A \cap B)$ = probability of both $A$ and $B$ .
Union (probability)
Examiner keyword
$P(A \cup B)$ = probability of $A$ or $B$ (or both).
Conditional probability
Examiner keyword
$P(A | B) = \dfrac{P(A \cap B)}{P(B)}$ . Probability of $A$ given $B$ has happened.

Common Mistakes and Misconceptions — Venn Diagrams

The traps other students keep falling into on venn diagrams questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Filling 'A only' region with $|A|$ instead of $|A| - |A \cap B|$ .
Why it happens
Confusing label with region.
How to avoid it
' $A$ ' includes the overlap. ' $A$ only' is the part of $A$ outside $B$ . Subtract the intersection.
✕Skipping the 'neither' region.
Why it happens
Focus on the circles.
How to avoid it
The 'neither' region is outside the circles. Always include it; the four regions sum to the total.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Venn Diagrams — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Statistics and ProbabilityIB MYP Maths Extended Level

Venn Diagrams

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next.

Previous Next

Learn this Topic

Start with the slides for the quick version, then go deeper with the full study notes.

Short Study Notes in the form of Slides

Read the notes first. If the method in a worked example clicks, you're ready for the questions.

Short Study Notes — Venn Diagrams

Start with these resources to cover the key concepts, then work through the practice questions.

Page 1 / 0

Detailed Notes

Full prose, callouts and a recap — built for A* mastery, not just a quick scan.

Take these study notes with you

Download a branded PDF — full prose, callouts, recap and memorise list for Venn Diagrams, ready to print or save offline.

Venn Diagrams (Probability) — IB MYP Mathematics (Extended): visualising events

Venn diagrams help compute probabilities for combined events. This MYP Mathematics Extended note covers two-set and three-set Venn diagrams used in probability.

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Fill in a Venn diagram from given probabilities or counts.
MYP Mathematics A — Compute probabilities of combined events.
MYP Mathematics C — Show clear Venn diagrams with all regions filled.

Two-set Venn diagram

Four regions to fill: A only, B only, both, neither.

A two-set Venn diagram has FOUR regions:

$A \cap B$ : in both.
$A$ only: in $A$ but not $B$ ( $A \setminus B$ ).
$B$ only: in $B$ but not $A$ .
Neither: outside both circles (but inside the universal set rectangle).

These four numbers add to the TOTAL (the universal set size).

Strategy for filling:

Start with $|A \cap B|$ (the intersection — often given).
Subtract from $|A|$ to get ' $A$ only'.
Subtract from $|B|$ to get ' $B$ only'.
Subtract everything from the total to get 'neither'.

Worked example. A class of 30 students: 18 like maths, 14 like art, 8 like both. Find $P(\text{likes only art})$ for a random student.

$A \cap B = 8$ .
Maths only: $18 - 8 = 10$ .
Art only: $14 - 8 = 6$ .
Neither: $30 - (10 + 8 + 6) = 6$ .
$P(\text{art only}) = 6/30 = 1/5$ .

Four regions in a two-set Venn.
Start from the centre (intersection), work outwards.
Sum of regions = total.

Three-set Venn diagram

Eight regions including 'neither'.

A three-set Venn has EIGHT regions (seven inside, one outside).

Strategy:

Start from $A \cap B \cap C$ (the centre).
Fill the pairwise overlaps that aren't in the centre.
Fill the 'only' regions.
Fill 'neither / outside'.

Worked example. Survey of 100 students:

50 play football
40 play basketball
30 play tennis
20 play football AND basketball
15 play football AND tennis
10 play basketball AND tennis
5 play all three.

Find how many play NONE.

Start from centre: $|F \cap B \cap T| = 5$ .

Pairwise (NOT in centre):

F and B only: $20 - 5 = 15$ .
F and T only: $15 - 5 = 10$ .
B and T only: $10 - 5 = 5$ .

Single-set 'only':

F only: $50 - (15 + 10 + 5) = 20$ .
B only: $40 - (15 + 5 + 5) = 15$ .
T only: $30 - (10 + 5 + 5) = 10$ .

Total inside at least one circle: $5 + 15 + 10 + 5 + 20 + 15 + 10 = 80$ .

Outside (none): $100 - 80 = 20$ .

So $P(\text{none}) = 20/100 = 1/5$ .

8 regions in a three-set Venn (7 inside + outside).
Always start from the centre (3-way intersection).
Work inside-out.

Using Venn diagrams for probability

Translate counts into probabilities.

Once the Venn is filled in, probability questions become 'count the regions you want' divided by total.

Common probability questions and the corresponding regions:

$P(A \cap B)$ : intersection (centre overlap).
$P(A \cup B)$ : everything inside either circle. (Equivalent: total $-$ neither.)
$P(A \setminus B)$ : $A$ only.
$P(A')$ : outside $A$ (which is $B$ only + neither).
$P(\text{exactly one}) = P(A \text{ only}) + P(B \text{ only})$ .

Worked example (continuing from above). What is $P(\text{at least two sports})$ ?

'At least two' = football+basketball-only ( $15$ ) + football+tennis-only ( $10$ ) + basketball+tennis-only ( $5$ ) + all three ( $5$ ).
Total: $15 + 10 + 5 + 5 = 35$ .
$P = 35/100 = 0.35$ .

A criterion-D question often gives a SURVEY data set and asks compound probability questions. Always:

Draw the Venn diagram.
Fill in ALL regions.
Read off the required count.
Divide by the total to get the probability.

Fill all regions FIRST.
Read off the count in the required region(s).
Divide by total to get probability.

How it’s examined

Criterion A: fill in Venn diagrams and compute probabilities. Criterion C: clear, labelled Venn diagrams. Criterion D: real survey data with compound probability questions.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

Master this Topic

Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

Take this whole topic with you

Download a branded revision sheet — worked examples, formulae, definitions and common mistakes for Venn Diagrams, ready to print or save as PDF.

Step-by-step worked examples — Venn Diagrams

Step-by-step solutions to past-paper-style questions on venn diagrams, written exactly the way a tutor would explain them at the board.

1Two-set Venn
Building confidence• two-set
Question
In a survey of 50 people: 30 like tea, 25 like coffee, 12 like both. (a) How many like tea only? (b) How many like neither?
Step-by-step solution
1. Step 1
  Both: 12. Tea only: $30 - 12 = 18$ .
2. Step 2
  Coffee only: $25 - 12 = 13$ .
3. Step 3
  Inside any circle: $18 + 12 + 13 = 43$ .
4. Step 4
  Neither: $50 - 43 = 7$ .
Answer
(a) 18 like tea only. (b) 7 like neither.
2Three-set Venn
Stretch• three-set
Question
$|A| = 25$ , $|B| = 30$ , $|C| = 20$ , $|A \cap B| = 10$ , $|A \cap C| = 8$ , $|B \cap C| = 12$ , $|A \cap B \cap C| = 5$ . Find the number in $A$ only.
Step-by-step solution
1. Step 1
  From $|A \cap B| = 10$ , the centre is 5, so 'A and B only' (not C) is $10 - 5 = 5$ .
2. Step 2
  From $|A \cap C| = 8$ , 'A and C only' = $8 - 5 = 3$ .
3. Step 3
  From $|A| = 25$ : A only = $25 - (5 + 3 + 5) = 25 - 13 = 12$ .
Answer
12 students are in $A$ only.
3Probability from Venn
Stretch• probability
Question
From the two-set Venn data above (50 people: 18 tea only, 12 both, 13 coffee only, 7 neither), find $P(\text{likes tea | likes coffee})$ .
Step-by-step solution
1. Step 1
  Conditional: restrict to coffee-likers. Total coffee-likers: $12 + 13 = 25$ .
2. Step 2
  Of those, $|T \cap C| = 12$ .
3. Step 3
  $P(\text{tea | coffee}) = 12/25$ .
Answer
$12/25 = 0.48$

Key Definitions and Keywords — Venn Diagrams

Definitions to memorise and the exact keywords mark schemes credit for venn diagrams answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Intersection (probability)
Examiner keyword
$P(A \cap B)$ = probability of both $A$ and $B$ .
Union (probability)
Examiner keyword
$P(A \cup B)$ = probability of $A$ or $B$ (or both).
Conditional probability
Examiner keyword
$P(A | B) = \dfrac{P(A \cap B)}{P(B)}$ . Probability of $A$ given $B$ has happened.

Common Mistakes and Misconceptions — Venn Diagrams

The traps other students keep falling into on venn diagrams questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Filling 'A only' region with $|A|$ instead of $|A| - |A \cap B|$ .
Why it happens
Confusing label with region.
How to avoid it
' $A$ ' includes the overlap. ' $A$ only' is the part of $A$ outside $B$ . Subtract the intersection.
✕Skipping the 'neither' region.
Why it happens
Focus on the circles.
How to avoid it
The 'neither' region is outside the circles. Always include it; the four regions sum to the total.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Venn Diagrams — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Venn Diagrams

Short Study Notes in the form of Slides

Short Study Notes — Venn Diagrams

Detailed Notes

What you’ll learn

How it’s examined

1Two-set Venn

2Three-set Venn

3Probability from Venn

Intersection (probability)

Union (probability)

Conditional probability

✕Filling 'A only' region with ∣A∣|A|∣A∣ instead of ∣A∣−∣A∩B∣|A| - |A \cap B|∣A∣−∣A∩B∣.

✕Skipping the 'neither' region.

Practice questions

Past paper style quiz

Assess your understanding

Venn Diagrams — frequently asked questions

Venn Diagrams

Short Study Notes in the form of Slides

Short Study Notes — Venn Diagrams

Detailed Notes

What you’ll learn

How it’s examined

1Two-set Venn

2Three-set Venn

3Probability from Venn

Intersection (probability)

Union (probability)

Conditional probability

✕Filling 'A only' region with ∣A∣|A|∣A∣ instead of ∣A∣−∣A∩B∣|A| - |A \cap B|∣A∣−∣A∩B∣.

✕Skipping the 'neither' region.

Practice questions

Past paper style quiz

Assess your understanding

Venn Diagrams — frequently asked questions

✕Filling 'A only' region with $|A|$ instead of $|A| - |A \cap B|$ .

✕Filling 'A only' region with $|A|$ instead of $|A| - |A \cap B|$ .