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.
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
Avoid These
The most common mistakes students make.
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 parents and students ask.
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