Probability

Venn Diagram Maker

Enter how big each set is and how much they overlap, and see the Venn diagram with the count in every region, plus the union and each probability.

Your sets

Total in the universe (N)

Everyone or everything being counted, including items in no set.

Name of set A

Size of Maths

Name of set B

Size of Physics

Name of set C

Size of Chemistry

Overlaps (including any in all sets)

Maths ∩ Physics

Maths ∩ Chemistry

Physics ∩ Chemistry

All three

Venn diagram

|A ∪ B ∪ C| = (60 + 45 + 30) − (25 + 15 + 12) + 8 = 91 — so 9 of the 100 are in none of the sets.

Probability	Working	Value
P(Maths)	60 ÷ 100	0.6
P(Physics)	45 ÷ 100	0.45
P(Chemistry)	30 ÷ 100	0.3
P(at least one)	91 ÷ 100	0.91
P(none)	9 ÷ 100	0.09
P(exactly one)	55 ÷ 100	0.55
P(exactly two)	28 ÷ 100	0.28
P(all three)	8 ÷ 100	0.08

How to use this tool

Choose two or three sets and type in the total number of items being counted.
Enter the size of each set, then the overlaps. An overlap such as A ∩ B includes the items that are also in the third set.
Read the count in each region of the diagram and the probabilities in the table below it.
If the numbers are impossible, for example an overlap larger than a set, the tool says which region goes negative.

Formula Reference

Union of two sets

|A ∪ B| = |A| + |B| − |A ∩ B|

Union of three sets

|A∪B∪C| = ΣA − Σ(pairs) + |A∩B∩C|

Region "only A" (3 sets)

|A| − |A∩B| − |A∩C| + |A∩B∩C|

Conditional probability

P(A | B) = |A ∩ B| ÷ |B|

ArticleThe Inclusion–Exclusion Principle ArticleJoint, Marginal & Conditional Probability ArticleBasic Probability CalcBasic Probability Calculator GlossaryGlossary: Mutually Exclusive Events

Frequently Asked Questions

What does the overlap box mean when I use three sets?+

Each pairwise overlap counts every item that is in both sets, including those that are also in the third set. The "all three" box is then counted inside each pairwise overlap. The tool subtracts it to find the region belonging to exactly two sets.

Why does the tool say a region would be negative?+

A region count cannot be below zero. If the overlaps you entered are larger than the sets allow, for example an A ∩ B larger than A, no real collection of items could match those numbers. Reduce the overlap or enlarge the set.

Are the circle sizes drawn to scale?+

No. The diagram is a standard three-circle layout with the counts printed in each region. Circle areas do not change with the numbers.

How do I check whether two events are independent?+

Compare P(A and B) with P(A) × P(B). If they are equal, the events are independent. With two sets selected, the tool does this check for you.

Probability

Working

Value

P(Maths)

60 ÷ 100

0.6

P(Physics)

45 ÷ 100

0.45

P(Chemistry)

30 ÷ 100

0.3

P(at least one)

91 ÷ 100

0.91

P(none)

9 ÷ 100

0.09

P(exactly one)

55 ÷ 100

0.55

P(exactly two)

28 ÷ 100

0.28

P(all three)

8 ÷ 100

0.08

How to use this tool

Choose two or three sets and type in the total number of items being counted.

Enter the size of each set, then the overlaps. An overlap such as A ∩ B includes the items that are also in the third set.

Read the count in each region of the diagram and the probabilities in the table below it.

If the numbers are impossible, for example an overlap larger than a set, the tool says which region goes negative.