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🎯 Conditional Probability and Independence Explained

A 3D scatter of sample-space outcomes inside two overlapping event regions: reshape P(A), P(B) and their dependence, then zoom into B to see conditional probability P(A|B) emerge live.

AI & Machine Learning3DAdvanced60 FPS

A 3D scatter of sample-space outcomes sits beneath two overlapping event discs, A and B. Reshaping their size and overlap — and zooming into B — turns conditional probability and independence from formulas into something you can literally count and watch shrink.

🔬 What It Demonstrates

Each dot is an outcome; a region's measured probability is just the fraction of dots inside it. P(A∩B) versus P(A)×P(B) tells you whether A and B are independent, and conditioning on B — shrinking every point outside it — makes P(A|B) visible as the share of B that's also A.

🎮 How to Use

Set P(A) and P(B) to resize the two discs, then drag Dependence from −1 (mutually exclusive) through 0 (independent) to +1 (nested). Toggle "Condition on B" to zoom the picture down to the world where B has already happened.

💡 Did You Know?

Independence is a numerical coincidence of a particular pair of probabilities, not a promise about cause and effect — two causally linked events can still satisfy P(A∩B)=P(A)P(B) for one specific setting, then stop being independent the moment either probability changes.

⚙ Under the hood

A 3D scatter of sample-space outcomes inside two overlapping event regions: reshape P(A), P(B) and their dependence, then zoom into B to see conditional probability P(A|B) emerge live.

Machine LearningProbabilityConditional ProbabilityIndependenceData AnalysisStatistical ModelsThree.js

3D · Three.js / WebGL renderer · 60 FPS target · runs fully client-side, no install