# Independence

In probability theory, two events are **independent**, **statistically independent**, or **stochastically independent**^{[1]} if the occurrence of one does not affect the probability of the other.

The concept of independence extends to dealing with collections of more than two events, in which case the events are pairwise independent if each pair are independent of each other, and the events are mutually independent if each event is independent of each other combination of events.

## Two events

Two events [math]A[/math] and [math]B[/math] are **independent** (often written as [math]A \perp B[/math] or [math]A \perp\!\!\!\perp B[/math]) if their joint probability equals the product of their probabilities:

Why this defines independence is made clear by rewriting with conditional probabilities:

and similarly

Thus, the occurrence of [math]B[/math] does not affect the probability of [math]A[/math], and vice versa. Although the derived expressions may seem more intuitive, they are not the preferred definition, as the conditional probabilities may be undefined if [math]\operatorname{P}([/math][math]A[/math]) or [math]\operatorname{P}([/math][math]B[/math]) are 0. Furthermore, the preferred definition makes clear by symmetry that when [math]A[/math] is independent of [math]B[/math], [math]B[/math] is also independent of [math]A[/math].

## More than two events

A finite set of events [math]A_i[/math] is **pairwise independent** if and only if every pair of events is independent^{[2]}—that is, if and only if for all distinct pairs of indices [math]m, k[/math]

A finite set of events is **mutually independent** if and only if every event is independent of any intersection of the other events^{[2]}—that is, if and only if for every [math]n[/math]-element subset [math]A_i[/math],

This is called the *multiplication rule* for independent events.

## Examples

### Rolling a die

The event of getting a 6 the first time a die is rolled and the event of getting a 6 the second time are *independent*. By contrast, the event of getting a 6 the first time a die is rolled and the event that the sum of the numbers seen on the first and second trials is 8 are *not* independent.

### Drawing cards

If two cards are drawn *with* replacement from a deck of cards, the event of drawing a red card on the first trial and that of drawing a red card on the second trial are *independent*. By contrast, if two cards are drawn *without* replacement from a deck of cards, the event of drawing a red card on the first trial and that of drawing a red card on the second trial are again *not* independent.

## Notes

- Russell, Stuart; Norvig, Peter (2002).
*Artificial Intelligence: A Modern Approach*. Prentice Hall. p. 478. ISBN 0-13-790395-2. -
^{2.0}^{2.1}Feller, W (1971). "Stochastic Independence".*An Introduction to Probability Theory and Its Applications*. Wiley.

## References

- Wikipedia contributors. "Independence (probability theory)".
*Wikipedia*. Wikipedia. Retrieved 28 January 2022.