Venn Diagram Visualization

The sample space Ω is partitioned into compartments. Event A (indigo ellipse) overlaps these compartments differently. Click on any compartment to see P(A | Bᵢ) - the probability of A given that specific compartment.

Click on any compartment to see the conditional probability

How This Visualization Works

Interactive Elements: The diagram shows a sample space Ω divided into compartments (B₁, B₂, B₃, etc.). Event A is represented by an indigo ellipse that overlaps multiple compartments. When you click on a compartment, the visualization highlights only the portion of A that intersects with that compartment, showing you the region A ∩ Bᵢ.

Conditional Probability Definition: The conditional probability P(A | Bᵢ) asks: Given that we know event Bᵢ has occurred, what is the probability that A also occurs? This is calculated as P(A | Bᵢ) = P(A ∩ Bᵢ) / P(Bᵢ). When we condition on Bᵢ, we restrict our view to only that compartment, treating it as our new sample space.

Why Probabilities Differ: Notice how P(A | B₁) ≠ P(A | B₂) ≠ P(A | B₃). This happens because event A overlaps each compartment to different degrees. Compartments where A has more overlap will have higher conditional probabilities. This demonstrates that knowing which compartment we are in (the condition) significantly affects the probability of event A.

Law of Total Probability: The total probability P(A) can be computed by summing the contributions from each compartment: P(A) = P(B₁)·P(A|B₁) + P(B₂)·P(A|B₂) + P(B₃)·P(A|B₃) + ... This formula shows that the overall probability of A is a weighted average of its conditional probabilities across all possible conditions (compartments), where each weight is the probability of that compartment.

Total Probability

P(A)

0.68

How Area(A) is calculated:

Area(A∩B₁) = 18
Area(A∩B₂) = 32
Area(A∩B₃) = 18
Area(A) = 18 + 32 + 18
= 68

P(A) from total area:

Area(A) = 68
Area(Ω) = 100
P(A) = 68 / 100
= 0.68

Verification (Law of Total Probability):

P(B₁)·P(A|B₁) = 0.33·0.54 = 0.18
P(B₂)·P(A|B₂) = 0.33·0.96 = 0.32
P(B₃)·P(A|B₃) = 0.33·0.54 = 0.18
Sum = 0.18 + 0.32 + 0.18
= 0.68

Settings

Number of compartments: 3

Conditional Probabilities

P(A | B₁)0.54

Step 1: Areas
Area(A ∩ B₁) = 18
Area(B₁) = 33.33
Area(Ω) = 100
Step 2: Convert to probabilities
P(A ∩ B₁) = 18/100 = 0.18
P(B₁) = 33.33/100 = 0.33
Step 3: Calculate conditional
P(A|B₁) = P(A∩B₁)/P(B₁)
= 0.18 / 0.33 = 0.54

P(A | B₂)0.96

Step 1: Areas
Area(A ∩ B₂) = 32
Area(B₂) = 33.33
Area(Ω) = 100
Step 2: Convert to probabilities
P(A ∩ B₂) = 32/100 = 0.32
P(B₂) = 33.33/100 = 0.33
Step 3: Calculate conditional
P(A|B₂) = P(A∩B₂)/P(B₂)
= 0.32 / 0.33 = 0.96

P(A | B₃)0.54

Step 1: Areas
Area(A ∩ B₃) = 18
Area(B₃) = 33.33
Area(Ω) = 100
Step 2: Convert to probabilities
P(A ∩ B₃) = 18/100 = 0.18
P(B₃) = 33.33/100 = 0.33
Step 3: Calculate conditional
P(A|B₃) = P(A∩B₃)/P(B₃)
= 0.18 / 0.33 = 0.54

Law of Total Probability

P(A) = 
P(B₁)·P(A|B₁) + 
P(B₂)·P(A|B₂) + 
P(B₃)·P(A|B₃)

Visualizing P(A|B)

Set Operations and Conditioning

Common Applications

Understanding Conditional Probability with Venn Diagrams

Venn diagrams show conditional probability through overlapping sets. When calculating P(A|B), we focus only on the region where B occurs and find what portion also contains A. This spatial representation makes conditional probability intuitive.

Visualizing P(A|B)

In a Venn diagram, P(A|B) is visualized by focusing on circle B and finding what fraction of it overlaps with A. The condition 'given B' means we're only looking within B's region.

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Set Operations and Conditioning

Venn diagrams naturally show how conditioning restricts the sample space. The overlap (intersection) divided by the condition set gives the conditional probability: P(A|B) = P(A ∩ B) / P(B).

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Common Applications

Venn diagrams are excellent for understanding medical test accuracy, survey data analysis, and any scenario involving overlapping categories or characteristics.

Learn More