Reading Venn Diagrams
Grade 4What a Venn diagram shows
A Venn diagram uses overlapping circles inside a rectangle to organise things into groups. Each circle is a set — a collection of items that share a property.
The overlap (the lens-shaped middle) contains items that belong to both sets. Items outside all circles but still inside the rectangle belong to neither set. The rectangle itself represents everything being considered — this is called the universal set, labelled ξ (or sometimes U or ε).
Filling in a Venn diagram
EXAMPLE
In a class of 30 students, 18 study French, 12 study Spanish, and 5 study both. How many study neither? Fill in the Venn diagram and find P(randomly chosen student studies French only).
Start with the overlap: 5 study both — write 5 in the middle section.
French only: 18 − 5 = 13. Spanish only: 12 − 5 = 7.
Total accounted for: 13 + 5 + 7 = 25. Neither: 30 − 25 = 5.
P(French only) = 13/30.
Every number belongs in exactly one region
When filling in a Venn diagram, always start with the overlap and work outwards. If you place the full group total in the circle first and then try to subtract, you'll count overlapping students twice.
Reading probabilities off the diagram
Once the diagram is filled in, every probability question becomes a reading exercise. To find the probability of a randomly chosen person being in a region, count the number in that region and divide by the total in the universal set.
The numbers in all regions of the diagram must add up to the total in the universal set. Use this as a check — if they don't add up, you've misplaced a value.
