Venn diagrams organise overlapping events so you can read off probabilities. Grade 11 questions use two or three events, often with an unknown region that you find with an equation.
Venn diagrams with two and three events
Fill a Venn diagram from the inside out:
- The region where all events overlap.
- The regions for exactly two events.
- The "only" regions for each event.
- The region outside all circles.
Two events. 50 learners: 28 in , 20 in and 8 in both. Then only = 20, only = 12, and outside = .
Three events. 100 learners: 45 play soccer, 30 netball, 25 chess. 10 play soccer and netball, 8 soccer and chess, 5 netball and chess, and 3 all three.
- All three: 3
- Exactly two: , ,
- Only soccer: ; only netball: ; only chess:
- None:
Finding unknowns in Venn diagrams
- Label the unknown region with a variable.
- Write every other region in terms of it.
- Use the total (number or probability 1) to form an equation.
Example. 50 learners: 25 in the Maths club, 20 in the Science club. are in both, and are in neither.
Check that every region is zero or positive. A negative region means an error.
Key terms
Venn diagram
A diagram of overlapping circles inside a rectangle representing the sample space.
Intersection
The overlap of events, ' and '.
Union
Everything in at least one event, ' or '.
Common mistakes
Filling in the given totals as 'only' regions.
Forgetting the region outside all circles.
Subtracting the triple overlap the wrong number of times in three-event diagrams.
Not checking that all regions add up to the total.
Practice questions
Reveal steps one at a time, then tick the marks you earned.
Question 1
4 marks total
In a group of 50 learners, 28 belong to the debating society (), 20 to the chess club (), and 8 belong to both.
Calculate the probability that a learner chosen at random belongs to neither.
Worked solution
Hidden - try the question first
Question 2
5 marks total
100 learners were surveyed. 45 play soccer (), 30 play netball () and 25 play chess (). 10 play and , 8 play and , 5 play and , and 3 play all three.
Draw a Venn diagram and calculate the probability that a learner plays none of the three.
Worked solution
Hidden - try the question first
Question 3
6 marks total
Use the survey in Question 2. Calculate the probability that a learner:
(a) plays chess only (2)
(b) plays exactly two of the activities (2)
(c) plays soccer or chess (2)
Worked solution
Hidden - try the question first
Question 4
8 marks total
In a group of 50 learners, 25 are in the Maths club () and 20 are in the Science club (). learners are in both, and are in neither.
(a) Calculate . (3)
(b) Calculate . (2)
(c) Are and independent? Show calculations. (3)
Worked solution
Hidden - try the question first
FAQs
Why fill a Venn diagram from the middle?
Each given number for an overlap includes the centre. Starting in the middle stops you counting people twice.
What if a region comes out negative?
Something is wrong. Check the subtractions and the order you filled the regions in.
Can I use a Venn diagram to test independence?
Yes. Read , and from the diagram and compare.