Probability
Grade 5-7Venn Diagrams & Set Notation
A Venn diagram sorts every member of a group into regions based on which sets they belong to, and set notation gives a precise shorthand for describing those regions. This lesson covers reading a Venn diagram, finding a missing value, using union, intersection and complement notation, and finding probabilities from a Venn diagram.
What you need to know
- The overlap is "A and B" (the intersection); everything inside either circle is "A or B" (the union).
- Anything outside both circles is in neither set.
- Fill the intersection first, then the rest of each circle, then the region outside.
- P(A given B) uses only the part of the diagram inside B.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Reading a Venn diagram
A Venn diagram uses overlapping circles to sort a group of items by which sets they belong to. The overlapping region shows items in both sets, the non-overlapping parts of each circle show items in only that set, and the space outside both circles shows items in neither set. The number in the corner, ξ, is the total number of items.
If the overlapping region between Football and Tennis shows 5, that means 5 students play both sports.
Finding a missing value
Every region in the diagram must add up to the total, ξ. Find a missing region by subtracting all the other known regions from the total.
With ξ = 30, and regions of 10, 6 and 9 already shown, the missing region outside both circles is 30 - 10 - 6 - 9 = 5.
Union, intersection and complement notation
The union symbol, ∪, means "or", and includes every item in either set, found by adding all the regions that belong to at least one of the sets. The intersection symbol, ∩, means "and", and only includes the overlapping region. A dash after a set, like A', means "not A", the complement, covering everything outside that set.
For sets F and T, n(F ∪ T) means the total number of students in Football or Tennis (or both), found by adding every region inside either circle. n((F ∩ T)') means everyone who is not in both Football and Tennis, which is everyone except the overlapping region.
Finding probabilities from a Venn diagram
To find the probability that a randomly selected item belongs to a particular region or combination of regions, divide the total in that region (or regions) by the overall total, ξ.
With ξ = 40, and 6 + 9 = 15 students in the Science club (including the overlap), the probability of picking a Science club student is 15/40, which simplifies to 3/8.
Worked examples
Three exam-style questions, fully solved
The Venn diagram shows the number of students who play Football (F) and Tennis (T), with ξ = 28: 12 play only Football, 5 play both, 8 play only Tennis, and 3 play neither. How many students play both Football and Tennis?
Easy- 1.Identify the overlapping region of the two circles
Answer: 5
Using the same Venn diagram (12 only Football, 5 both, 8 only Tennis, 3 neither), find n(F ∪ T).
Medium- 1.Recognise that F ∪ T means everyone in Football or Tennis, or both
- 2.Add every region inside either circle: 12 + 5 + 8
Answer: 25
The Venn diagram shows information about 20 students: (x + 2) play only Football, 5 play both, x play only Tennis, and 3 play neither. Find the value of x.
Hard- 1.Write an equation using all four regions adding to the total: (x + 2) + 5 + x + 3 = 20
- 2.Simplify: 2x + 10 = 20
- 3.Solve for x: 2x = 10
Answer: x = 5
Practice
6 questions, marked instantly
Type an answer and check it. The worked solution appears once you have had a go. No login needed, and your progress saves in this browser.
Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
In a Venn diagram, set A has 12 members, set B has 9 members, and 4 members are in both A and B. How many members are in A but not B?
Using the same Venn diagram (A has 12, B has 9, 4 in both), how many members are in A or B (A ∪ B) altogether?
30 students were asked. 18 study French, 14 study German, and 6 study both. How many study neither language?
A Venn diagram of sets A and B has: 7 in A only, 5 in B only, 3 in both, and 2 in neither. How many members are there altogether?
Using that Venn diagram (7 in A only, 5 in B only, 3 in both, 2 in neither), a member is picked at random. Work out the probability they are in set A.
Using that same Venn diagram, work out P(A ∩ B), the probability a member is in both sets.
Avoid these
The mistakes students make most often
Confusing the union symbol, ∪, meaning "or" (everything in either set), with the intersection symbol, ∩, meaning "and" (only the overlap).
Forgetting to include the overlapping region when finding a union, or accidentally counting it twice.
Misreading the complement symbol, ', and including the set itself instead of everything outside it.
When finding a missing value, forgetting to subtract every other known region, including the number outside both circles.
When solving for x using algebraic regions, forgetting to include the region outside both circles in the total equation.
FAQ
Questions students and parents ask
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