A Venn diagram uses overlapping circles inside a rectangle to show how items belong to groups. The overlap holds items in both groups, each crescent holds items in one group only, and the space outside the circles holds items in neither. To fill one in from totals, start with the overlap, then subtract to find each “only” region, and finally the “neither” region. Probabilities are a region's count divided by the total.
Venn diagram regions
- Overlap: in both groups
- A only = total in A − overlap
- Neither = overall total − everything inside the circles
- P(region) = number in the region ÷ total
The regions of a Venn diagram
A two-circle Venn diagram has four regions: the overlap (both), the part of each circle outside the overlap (one group only), and the space outside both circles (neither). The rectangle represents everyone or everything in the question.
Filling in from totals
Always start with the overlap. If 34 people swim, 25 cycle and 9 do both, put 9 in the overlap, then 34 − 9 = 25 in “swim only” and 25 − 9 = 16 in “cycle only”. Finally, subtract everything inside the circles from the total to get “neither”: with 60 people, 60 − 50 = 10.
Reading a Venn diagram
To find how many are in a whole group, add both parts of its circle, the “only” region and the overlap. “Exactly one” means the two “only” regions added together, not including the overlap. Read each question's wording carefully.
Probabilities from a Venn diagram
Divide the number in the region you want by the total. With the swimming and cycling diagram, P(neither) = 10/60 = 1/6 and P(exactly one activity) = (25 + 16)/60 = 41/60.
Working backwards
Some questions give the regions and ask for a missing total, or give the probability of a region and ask for a missing value. Use the fact that the regions add up to the total to form an equation.
Worked examples
Example 1. Of 60 people, 34 swim, 25 cycle and 9 do both. Complete a Venn diagram.
- Overlap: 9.
- Swim only: 34 − 9 = 25. Cycle only: 25 − 9 = 16.
- Neither: 60 − (25 + 9 + 16) = 10.
Answer: Swim only 25, both 9, cycle only 16, neither 10.
Example 2. Using the diagram, find P(a person does neither activity).
- 10 out of 60.
Answer: 10/60 = 1/6.
Example 3. Find P(a person does exactly one of the activities).
- Swim only + cycle only: 25 + 16 = 41.
Answer: 41/60.
Example 4. How many people swim in total, and how many do not swim?
- Swim in total: 25 + 9 = 34.
- Do not swim: 60 − 34 = 26.
Answer: 34 swim; 26 do not.
Example 5. In a class, 13 study only art, x study both art and drama, 9 study only drama, and 4 study neither. There are 32 students. Find x.
- 13 + x + 9 + 4 = 32, so 26 + x = 32.
Answer: x = 6.
Common mistakes
- Putting the full total in a circle's crescent. Subtract the overlap first: “only” = total − both.
- Forgetting the “neither” region. Subtract everything in the circles from the overall total.
- Including the overlap in “exactly one”. Exactly one means the two “only” regions.
- Using the wrong denominator. Divide by the overall total unless a group is specified.
- Starting from the outside. Always start filling in from the overlap.
How Venn diagrams are examined
Venn diagrams appear on Foundation papers for 1MA1 and 4MA1, and at Higher tier they are combined with set notation and conditional probability. Questions ask you to:
- complete a Venn diagram from given totals
- find the number in a region or group
- find probabilities from a Venn diagram
- find a missing value using the total
Practise with the Venn diagrams worksheet, which includes answers, and the probability engine.
At Higher tier, Venn diagrams use set notation. Compare them with two-way tables.
Frequently asked questions
How do you fill in a Venn diagram?
Start with the overlap, subtract it from each group's total for the “only” regions, then subtract everything from the overall total for “neither”.
What does the overlap of a Venn diagram show?
The items that belong to both groups.
How do you find a probability from a Venn diagram?
Divide the number in the region by the total number of items.