The 3-set Venn diagram displays three overlapping circles labeled A, B, and C. Eight numbered segments represent all possible outcome combinations from three events.
The center segment (#1) shows the triple intersection where all three events occur simultaneously. Segments #2-#7 represent various two-way intersections and single-event-only regions. Segment #8 lies outside all circles.
View the pre-loaded "Demographics Study" example showing how constraints like P(A∩Bc)=0.4 and P(A∩Cc)=0.18 determine all eight region probabilities through systematic calculation.
Navigating Eight Regions
Click any numbered segment (1-8) to select and highlight it. The corresponding outcome displays in the middle panel with its probability value and explanation.
Segments #1-7 lie within at least one circle. Segment #8 represents Ac∩Bc∩Cc - outcomes where none of the three events occur.
Hover over segments to preview which outcome they represent. Hover over outcomes in the list to see their diagram location. This bidirectional connection clarifies complex notation like A∩Bc∩C.
Understanding the Eight Outcomes
Segment #1 (A∩B∩C): All three events occur - the center intersection.
Segments #2-4: Two events occur, one doesn't - the three two-way intersections excluding the center.
Segments #5-7: Only one event occurs - portions of single circles that don't overlap with others.
Segment #8 (Ac∩Bc∩Cc): None of the events occur - outside all three circles.
These eight mutually exclusive regions partition the entire sample space, and their probabilities sum to exactly 1.000.
Using Show Calculations
Toggle "Show Calculations" to reveal step-by-step solutions for all eight regions. The tool displays the logical sequence for solving the system of probability equations.
The solution typically starts by finding pairwise intersections: P(A∩B)=P(A)−P(A∩Bc). Then it solves for the triple intersection using all available constraints.
Once the triple intersection is known, other regions follow systematically. For example, P(A∩B∩Cc)=P(A∩B)−P(A∩B∩C). Each calculation builds on previous results.
Reading Probability Solutions
Each of the eight outcomes displays its calculated probability to three decimal places. In the Demographics example, P(A∩B∩C)=0.080 means 8% of the population are women, unemployed, and have academic background.
Compare region sizes to understand relationships. If P(A∩Bc∩C)=0.240 is much larger than P(A∩B∩C)=0.080, academic women are predominantly employed in this dataset.
The probabilities reflect both the given marginals P(A),P(B),P(C) and the specific constraints. Different constraints yield completely different regional distributions.
Solving Systems of Equations
Three-set problems require solving systems of equations. Given three marginals and two or three constraints, the tool determines all eight unknown region probabilities.
The general approach: (1) Use constraints to find pairwise intersections like P(A∩B). (2) Set up equations relating the triple intersection to known values. (3) Solve for the center region. (4) Calculate remaining regions systematically.
Click "Show Calculations" to see the General Solution Steps outlining this process. Individual region calculations appear when you select specific segments.
The Inclusion-Exclusion Principle
For three sets, the inclusion-exclusion principle calculates P(A∪B∪C) - the probability that at least one event occurs.
The formula adds individual probabilities, subtracts pairwise overlaps (to correct double-counting), then adds back the triple intersection (which was subtracted three times but should only be subtracted twice).
In the diagram, P(A∪B∪C) equals segments #1 through #7 - everything inside at least one circle.
Conditional Probability with Three Events
Conditional probability extends to three events. P(A∩B∣C) asks: given C occurred, what's the probability both A and B occur?
Visually, restrict to circle C (segments #1, #3, #5, #7), then find what fraction also lies in both A and B (only segment #1).
Calculate as P(A∩B∣C)=P(C)P(A∩B∩C). If P(A∩B∩C)=0.08 and P(C)=0.62, then P(A∩B∣C)=0.08/0.62≈0.129 or 12.9%.
When to Use 3-Set Diagrams
Use 3-set Venn diagrams when analyzing three distinct characteristics or categories simultaneously. Common applications include:
Survey analysis: Demographic studies with three attributes (age group, employment status, education level)
Medical research: Patients with three conditions or risk factors
Quality control: Products with three types of potential defects
Market segmentation: Customers categorized by three preferences or behaviors
For four or more events, contingency tables or tree diagrams become more practical than Venn diagrams.