Venn Diagrams and Counting Outcomes
Construct and interpret Venn diagrams with two or three sets to organise and count outcomes; use systematic listing and the product rule for counting to enumerate all possible outcomes of combined events
What a learner can do afterwards
- Draw a two-circle Venn diagram to sort 30 students by whether they like football, like cricket, or like both
- Shade the intersection A ∩ B and the union A ∪ B on a Venn diagram and explain what each region represents
- Use a completed Venn diagram to calculate P(A), P(B), P(A ∩ B), and P(A ∪ B)
1 · Read
You've already drawn Venn diagrams to sort things into groups. Now you'll use the same diagram to count real groups of people, like students who play different sports, and to work out how likely you are to pick one of them at random.
A class of 30 students is asked about football and cricket. 18 like football, 15 like cricket, and 8 like both. The 8 who like both go in the middle, the intersection. Only football is 18 minus 8, which is 10. Only cricket is 15 minus 8, which is 7. Adding 10 + 7 + 8 gives 25 students who like at least one sport, so 30 minus 25 leaves 5 who like neither. The probability of picking a student who likes both sports is 8 out of 30, so P(A ∩ B) = 8/30.
When two circles overlap, count the middle part once, not twice. Add only-A, only-B, and both together to get the total for A or B.
The intersection is the overlap counted once, the union is everyone in A, B, or both, and a region's count divided by the total gives its probability.
More to read
Probability and combinatorics: FAQ
A Venn diagram draws sets as overlapping circles. Each circle holds one group: football fans left, cricket fans right. The shared lens in the middle is the intersection: fans of both. Everything inside either circle is the union: fans of at least one. The box around holds everyone surveyed; outsiders like neither. Fill the overlap first (8), then the crescents (10 football-only, 6 cricket-only), then the outside (6 neither). Probabilities divide by the class total: P(football) = 18/30 = 3/5, P(both) = 8/30 = 4/15, P(either) = 24/30 = 4/5.
3.5 Venn Diagrams
A Venn diagram pictures experiment outcomes. The box is the sample space: all 30 students. Circles are events: likes football, likes cricket. Overlaps show shared outcomes. English maps to regions: both means the overlap, either means the union, neither means outside everything. Counting flows outward: overlap first, crescents next, outside last. With 10, 8, 6, 6 placed, every question answers itself: only-football is 10, neither is 6, and each probability divides its region by 30.
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2 · Watch
به انگلیسیTake it off screen
Where it sits
Learn first
Nothing; this is a starting point.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
16 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.