PROBABILITY
The invisible architecture of chance, made visible. Where randomness resolves into order, and uncertainty crystallizes into understanding.
The fundamental measure: the ratio of favorable outcomes to all possible outcomes.
When we know something has already happened, how does it reshape what we expect next? Conditional probability is the mathematics of updated belief -- the formal machinery behind intuition.
Bayes' foundation: the probability of A given B equals their joint probability divided by the probability of B.
Drag the circles in the visualization to explore how overlap changes conditional probability. The intersection represents events that satisfy both conditions simultaneously.
Bayes' theorem: reversing the conditional. The engine of modern machine learning and medical diagnostics.
The bell curve emerges everywhere: heights, test scores, measurement errors, thermal fluctuations. The central limit theorem guarantees that the sum of many independent random variables tends toward this shape, regardless of their individual distributions.
The probability density function. Two parameters -- mean (μ) and standard deviation (σ) -- define the entire curve.
Drag left/right on the visualization to shift the mean. Drag up/down to change the standard deviation. Double-click to reset to the standard normal.
The expected value is the center; the variance measures the spread. Together they capture the essence of any distribution.
Humans see patterns everywhere, even in pure noise. This is both our greatest cognitive gift and our deepest statistical flaw. True randomness looks less random than we expect -- clusters form, gaps appear, and our pattern-seeking minds insist on meaning where none exists.
Clustering algorithms find structure even in randomness. The question is whether the structure is real or imposed.
Click the reshuffle button to scatter dots to new random positions and watch k-means re-cluster them. Each time, new "patterns" emerge from pure chance.