Venn Diagrams
Visualizing sets and calculating probabilities
Sample Space (S)
The rectangle contains all possible outcomes. Total probability within equals 1.
Intersection (A ∩ B)
The overlapping area represents outcomes belonging to both Event A AND Event B.
Union (A ∪ B)
All outcomes in A, B, or both. Used for "OR" probability calculations.
Complement (A')
Everything outside circle A — outcomes NOT in Event A. P(A') = 1 - P(A).
Mutually Exclusive
Circles that don't overlap — no common outcomes (A ∩ B = ∅).
Worked Example: Soccer & Rugby
Problem: In a class of 30 learners, 15 play soccer (A), 10 play rugby (B), and 4 play both sports. Find the probability a random learner plays only soccer.
- Step 1: Fill intersection first: Both = 4
- Step 2: Only Soccer = 15 - 4 = 11
- Step 3: Only Rugby = 10 - 4 = 6
- Step 4: Total playing sports = 11 + 6 + 4 = 21
- Step 5: Neither sport = 30 - 21 = 9
- Step 6: P(Only Soccer) = 11/30
Interactive Venn Diagram Builder
Adjust the numbers to see how the Venn diagram changes. Values update probabilities in real-time!
Probability Calculator
P(B) = 10/30 = 0.333
P(A ∩ B) = 4/30 = 0.133
P(A ∪ B) = 21/30 = 0.700
P(Only A) = 11/30 = 0.367
Set Operation Quiz
Test your knowledge of Venn diagram notation!
Test Your Understanding
Question 1: In a Venn diagram, what does the overlapping region of two circles represent
Question 2: If P(A) = 0.6 and P(A') is the complement, what is P(A')
Question 3: Two events that have no outcomes in common are called:
Key Takeaways
Key Terms
Exam Focus: Venn Diagrams
Always place the overlap first when filling in a Venn diagram. The intersection belongs to both sets, so it must be subtracted from the outside parts of each circle if the totals include the overlap. This is the mistake that most often leads to double counting.
After filling in the diagram, check that every region adds up to the total sample space. If the question includes values outside both circles, place them in the rectangle but outside the circles, because they are still part of the full group being studied.
Use labels such as only A, only B, both, and neither while working. These labels turn the diagram into a checklist and make probability questions easier to translate.
