Statistical independence defines the absence of any probabilistic relationship between two or more events.
Two events A and B are independent if and only if their joint probability equals the product of their individual probabilities.
Equivalently, knowing that event B occurred leaves the conditional probability of event A completely unchanged.
For example, flipping a coin twice produces independent trials because the physical mechanism retains no memory of past tosses.
Independence simplifies probability calculations by allowing joint likelihoods to be evaluated through direct multiplication.
In machine learning, the naive Bayes classifier assumes all input features are conditionally independent given the target class label.
Although this feature independence assumption is rarely true in real data, the resulting models often perform remarkably well.
Independence must not be confused with physical causality: two events can be correlated without either causing the other.
Conditional independence occurs when two events become independent only after conditioning on a third shared variable.
For example, shoe size and reading ability correlate strongly among children, but become independent once age is conditioned on.
In financial modelling, assuming default events across mortgage portfolios were independent led directly to the two thousand and eight financial crisis.
When systemic economic conditions deteriorated, defaults clustered together, violating the independence assumptions baked into pricing models.
Pairwise independence among three events does not guarantee mutual independence across all three events simultaneously.
Analysts must verify independence through empirical testing rather than assuming it as a matter of mathematical convenience.
A widespread misconception is that independent random processes must balance out past runs of bad luck, known as the gambler fallacy.