Hassan Ijaz
Ai, Web & Design
Conditional probability and independence
Two-box simulator where users can drag colored balls between boxes and see how P(A|B) changes dynamically with visual tree diagrams
Concept Overview
Conditional probability is the probability of an event occurring given that another event has already occurred. It's a fundamental concept for understanding how events relate to each other.
Key Concepts
- P(A|B): Probability of A given B has occurred = P(A ∩ B) / P(B)
- Forward Probability: P(Effect|Cause) - "If we know the cause, what's the probability of the effect?"
- Reverse Probability: P(Cause|Effect) - "If we observe the effect, what's the probability of the cause?"
Bayes' Theorem
P(A|B) = (P(B|A) × P(A))/ P(B)
This formula lets us "flip" conditional probabilities. If we know P(B|A), we can calculate P(A|B).
Real-World Example
Imagine two boxes of colored balls:
- Box 1: Mostly red balls
- Box 2: Mostly blue balls
Questions we can answer:
- If I pick from Box 1, what color will I likely get? (Forward)
- If I have a red ball, which box did it likely come from? (Reverse)
Use the interactive visualization below to drag balls between boxes and see how conditional probabilities change in real-time.
Interactive Visualization
Loading interactive visualization...
Two-box simulator where users can drag colored balls between boxes and see how P(A|B) changes dynamically with visual tree diagrams