Law of Total Probability | Probability Visualizer

P(B) = Σᵢ P(B|Aᵢ) · P(Aᵢ)

Sum over all mutually exclusive, exhaustive events Aᵢ

🌳 Probability Tree

📊 Partition of Sample Space

A₁

P(A₁) = 0.30

P(B|A₁) = 0.80

A₂

P(A₂) = 0.50

P(B|A₂) = 0.40

A₃

P(A₃) = 0.20

P(B|A₃) = 0.10

Adjust Prior Probabilities

P(A₁)

P(A₂)

P(A₃)

📐 Calculation

P(B|A₁)·P(A₁) 0.240

P(B|A₂)·P(A₂) 0.200

P(B|A₃)·P(A₃) 0.020

P(B) = Sum 0.460

Total Probability P(B)

0.460

💡 Key Insight

The law of total probability lets us calculate P(B) by considering how likely B is under each possible "cause" Aᵢ, weighted by how likely each cause is. This is the foundation for Bayes' theorem!