The 2-set Venn diagram displays two overlapping circles labeled A and B. Four numbered segments represent all possible outcomes: both events (intersection), A only, B only, and neither event (complement).
View the pre-loaded example problems by clicking the problem name buttons. Each problem shows event descriptions, marginal probabilities P(A) and P(B), and given constraints that help determine all four region probabilities.
The diagram automatically calculates probabilities for all four segments based on the given information. Click any numbered segment to highlight it and see detailed calculations.
Clicking and Hovering Interactions
Click any numbered segment (1-4) in the diagram to select it. Selected segments highlight in gold and display their corresponding outcome below. Click again to deselect.
Hover over segments to preview which outcome they represent. The segment highlights in yellow, and the matching outcome in the list also highlights.
Hover over outcomes in the left panel to see their location in the diagram. This two-way interaction helps connect symbolic notation like A∩B with visual diagram regions.
Understanding the Four Regions
Segment #1 ($A \cap B$): The intersection where both circles overlap. Both events occur simultaneously. This is the most central region of probability calculations.
Segment #2 ($A \cap B^c$): Event A occurs but B does not. This is the left portion of circle A that doesn't overlap with B.
Segment #3 ($A^c \cap B$): Event B occurs but A does not. This is the right portion of circle B that doesn't overlap with A.
Segment #4 ($A^c \cap B^c$): Neither event occurs. This region lies outside both circles, representing outcomes where both A and B fail to happen.
Using Show Calculations Feature
Toggle the "Show Calculations" button to reveal step-by-step solutions for each region's probability. The calculations explain how constraints and marginal probabilities determine each segment.
For segment #1, the calculation typically starts with given information like P(A∩B)=0.15. For segment #2, it uses P(A∩Bc)=P(A)−P(A∩B).
Each calculation shows the formula and the numerical result. This feature helps understand the logical sequence for solving Venn diagram problems from constraints to complete solutions.
Reading Probability Values
Each outcome displays its calculated probability on the right side, shown to three decimal places. These values always sum to exactly 1.000, representing the complete sample space.
The probabilities reflect the given constraints and marginal probabilities. In the "Student Survey" example, P(A∩B)=0.150 means 15% of the population are both students and employed.
Compare different regions to understand relative likelihoods. If P(A∩Bc)=0.450 is much larger than P(A∩B)=0.150, most students are unemployed in this scenario.
Working with Multiple Problems
Switch between example problems using the buttons at the top. Each problem has different event descriptions, marginal probabilities, and constraints.
The "Student Survey" problem examines students and employment. The "Health Screening" problem analyzes medical test accuracy with true positives and false positives.
Each problem demonstrates different constraint types. Some give P(A∩B) directly, others give P(A∩Bc), requiring different solution approaches.
Understanding Set Operations
Set operations form the mathematical foundation of Venn diagrams. The intersection A∩B represents "A and B" - outcomes where both events occur.
The union A∪B represents "A or B" - outcomes in either circle. Calculate it as P(A∪B)=P(A)+P(B)−P(A∩B), subtracting the overlap to avoid double-counting.
The complement Ac represents "not A" - everything outside circle A. By definition, P(A)+P(Ac)=1. These operations extend naturally to three or more sets.
Conditional Probability in 2-Set Diagrams
Conditional probabilityP(A∣B) asks "what's the probability of A given that B occurred?" Visually, focus only on circle B, then find what fraction of B overlaps with A.
Calculate it as P(A∣B)=P(B)P(A∩B). If P(A∩B)=0.15 and P(B)=0.40, then P(A∣B)=0.15/0.40=0.375 or 37.5%.
The diagram makes this intuitive: restrict attention to region B (segments #1 and #3), then see what portion is also in A (only segment #1).
Applying the Addition Rule
The addition rule calculates the probability of A or B occurring: P(A∪B)=P(A)+P(B)−P(A∩B). The diagram shows why subtraction is necessary.
Adding P(A) and P(B) counts the intersection twice - once in A's total and once in B's total. Subtracting P(A∩B) corrects this double-counting.
In terms of segments: P(A∪B) equals segments #1 + #2 + #3, which is P(A)+P(B)−P(A∩B) algebraically. The visual matches the formula perfectly.
Related Probability Tools
3-Set Venn Diagrams - Extend these concepts to three events with eight regions
Contingency Tables - Organize the same probability information in tabular format
Tree Diagrams - Show sequential conditional probabilities over time