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In [[ | In [[probability theory|probability theory]], two [[event (probability theory)|event]]s are '''independent''', '''statistically independent''', or '''stochastically independent'''<ref name="Artificial Intelligence">{{cite book | last = Russell| first =Stuart| last2 = Norvig | first2 = Peter | title = Artificial Intelligence: A Modern Approach | page = 478 | publisher = [[Prentice Hall|Prentice Hall]] | year = 2002 | isbn = 0-13-790395-2}}</ref> 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. | 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 == | ||
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 [[ | 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|joint probability]] equals the product of their probabilities:<math display="block">\mathrm{P}(A \cap B) = \mathrm{P}(A)\mathrm{P}(B).</math> | ||
Why this defines independence is made clear by rewriting with conditional probabilities:<math display="block"> | Why this defines independence is made clear by rewriting with conditional probabilities:<math display="block"> | ||
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== More than two events == | == More than two events == | ||
A finite set of events <math>A_i</math> is '''pairwise independent''' [[ | A finite set of events <math>A_i</math> is '''pairwise independent''' [[if and only if|if and only if]] every pair of events is independent<ref name ="Feller">{{cite book | last = Feller | first = W | year = 1971 | title = An Introduction to Probability Theory and Its Applications | publisher = [[John Wiley & Sons|Wiley]] | chapter = Stochastic Independence}}</ref>—that is, if and only if for all distinct pairs of indices <math>m, k</math> | ||
<math display="block">\mathrm{P}(A_m \cap A_k) = \mathrm{P}(A_m)\mathrm{P}(A_k).</math> | <math display="block">\mathrm{P}(A_m \cap A_k) = \mathrm{P}(A_m)\mathrm{P}(A_k).</math> |
Latest revision as of 15:14, 4 April 2024
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.