Concept/Probability and Statistics/No. 0503
Independence
Independence means that knowing one event occurred leaves the chance of another unchanged. In classical probability theory, two independent events have a joint chance equal to the product of their own chances. For larger groups, the product rule must hold for every subset.
Also called Statistical Independence · Probabilistic Independence
- Evidence
- Well established
- Read
- 6 min
- Links
- 13 connections
01You've seen this when…
- in life
You count two salaries as a financial safety net. Both jobs depend on the town’s largest employer.
- at work
A launch review counts five favorable customer interviews as five pieces of evidence. All five customers work for the same company and follow one purchasing policy.
- out in the world
Three news sites carry the same claim. Tracing their reports leads to a single wire story.
02The idea
Learning one thing often changes the chance of another. If the streets are wet, rain becomes more likely. If a company announces layoffs, an employee’s risk of losing their job rises. Independence describes the cases where learning one event occurred leaves the probability of another unchanged.
For two events, A and B, the mathematical rule is:
P(A and B) = P(A) × P(B).
Suppose two fair dice are rolled independently. Each has a 1-in-6 chance of showing a six. The chance that both show six is 1/6 × 1/6, or 1/36. Learning the first die’s result leaves the second die’s probabilities unchanged.
The same idea applies to measured quantities, such as delivery times or investment returns. Independence requires that information about one leaves the entire probability distribution of the other unchanged. Checking only their averages gives much less information.
The product rule also handles events with probability zero. The equivalent definition using conditional probability, P(B given A) = P(B), requires A to have positive probability.
03Why it matters
Independence lets people combine separate probabilities into a probability for the whole situation. That makes it central to backup systems, forecasts, experiments and financial risk.
A second safeguard can greatly reduce risk when its failures are independent of the first. Shared vulnerabilities can erase much of that gain. A second witness adds less evidence when both witnesses heard the same rumor. Diversification provides less protection when investments depend on the same economic conditions.
Independence also affects how much information a sample contains. A thousand independent observations generally support more precise estimates than a thousand closely linked observations. Repeated measurements from one person, household or machine require attention to that dependence.
Dependence can change calculations in either direction. Shared causes often make joint failures more likely; events that exclude each other make joint occurrence impossible. The direction matters as much as the presence of a connection.
04A worked example
Imagine an online shop choosing a backup checkout host. The main host and backup are each unavailable during 1% of randomly sampled hours. These figures are invented for this example.
What it looks like A strong safety improvement. Multiplying 1% by 1% gives a 0.01% chance that both hosts are unavailable at a randomly chosen hour: about one hour in 10,000.
What’s actually going on Both hosts use the same upstream network. That network is down during 0.5% of hours, taking both hosts offline together. Their joint unavailability is therefore at least 0.5%, or about 50 hours in 10,000. The independence calculation understates it by at least a factor of 50.
The individual 1% figures can both be accurate. The error enters when the team multiplies them without checking how the failures overlap. This is a common-cause failure.
What would have helped Choose a backup with a different upstream network, then inspect other shared dependencies, including power, software and administrative access. Measure simultaneous outages directly where possible. Different providers reduce some shared risks, but their names alone cannot establish independence.
05Where people trip up
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Physical separation can hide shared causes. Two servers in different buildings can depend on the same power grid. Two studies can inherit the same measurement error. Trace what produces each outcome before treating them as independent. Confounding is one way a shared cause creates an association.
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Pairwise independence leaves a gap. Every pair in a group can be independent while the whole group is dependent. Toss two independent fair coins. Let A mean the first lands heads, B mean the second lands heads, and C mean they land on different sides. Each event has probability 1/2, and each pair occurs together with probability 1/4. Yet all three together are impossible. Multiplying their separate probabilities would give 1/8. Independence among several events requires the product rule for every subset, including the full group.
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Zero correlation is a weaker condition. Correlation usually measures a linear relationship. If X is equally likely to be −1, 0 or 1, and Y = X², their correlation is zero. Observing X still determines Y exactly. Independence implies zero correlation when the relevant moments exist; the reverse implication generally fails.
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Equal chances say little about dependence. A program can toss one fair coin and repeat its result forever. Every position in the resulting sequence has a 50% chance of heads, while observing the first result reveals all later results. Conversely, independent events can have different probabilities. Fairness, sameness of distribution and independence are separate properties.
Before multiplying probabilities, identify the exact events and the information already available. Look for shared inputs and feedback. For data collected over time, check whether neighboring observations move together through autocorrelation.
Independence also helps diagnose the gambler’s fallacy. Under an independent fair-coin model, a run of heads leaves the next toss’s chance of tails at 1/2.
06Where it doesn’t survive added information
Independence depends on the information being conditioned on. Two fair dice rolled independently become linked if someone reveals that their sum is seven. Learning that the first shows six then determines that the second shows one. Restricting attention to outcomes with a particular total has created dependence.
Conditioning can also remove an association. Heating use in two houses may move together because both respond to outdoor temperature. A model might treat their remaining variation as independent after accounting for temperature. That is a further assumption to examine.
Exact independence is a mathematical condition. In applications, approximate independence can be adequate for one purpose and dangerous for another. Small dependence may have little effect on a rough average yet substantially change the probability of several rare failures occurring together.
State the assumption alongside the calculation, and test how the decision changes when outcomes are more strongly linked. This makes the model’s vulnerability visible.
07Roots
In early eighteenth-century London, Abraham de Moivre worked on the mathematics of games of chance. A player rolling several dice needed the chance of a combined outcome, such as every die showing the same chosen face. Multiplying the separate chances supplied an answer when the dice behaved independently.
De Moivre’s The Doctrine of Chances, first published in 1718, helped organize these calculations. Its treatment distinguished independent events from dependent ones by whether one event altered the probability of another. Gambling gave the distinction practical force: drawing cards without replacement changes the deck, while separate dice rolls can preserve the original probabilities.
The idea developed within classical probability rather than through a single discovery. In 1933, Andrey Kolmogorov placed probability within a general axiomatic framework. Independence became a precise condition on joint probabilities, applicable to events and random quantities far beyond games. The same multiplication rule now supports calculations about samples, machine failures and streams of data.
08How solid is this?
Independence is a formally defined mathematical property, and the product rule and distinction between pairwise and joint independence are proven results. Whether particular events are independent is an empirical and modeling question; limited data cannot certify exact independence.
09Connections
- Often confused with Correlation
- Helps counter Gambler’s Fallacy
- In tension with Information Cascade
- Part of Conditional Probability, Diversification
- See also Bayes’ Theorem, Bayesian Updating, Hot Hand Fallacy, Common-Cause Failure, Autocorrelation, Confounding, Law of Large Numbers, Wisdom of Crowds
+ 3 more in the list
10Origin and sources
Developed within classical probability theory. Abraham de Moivre explained independent and dependent events in The Doctrine of Chances, first published in 1718. Andrey Kolmogorov incorporated independence into his axiomatic framework in 1933.
- [1]De Moivre, A. (1756). The Doctrine of Chances: or, A Method of Calculating the Probabilities of Events in Play. Third edition. A. Millar.
- [2]Kolmogorov, A. N. (1956). Foundations of the Theory of Probability. Second English edition. Chelsea Publishing Company.
- [3]Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Volume I. Third edition. John Wiley & Sons.
Suggest an edit· Updated 2026-10-02