Editing Independence (probability theory) (section)

====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>, where the latter symbol often is also used for [[conditional independence]]) if and only if their [[joint probability]] equals the product of their probabilities:<ref name=Florescu>{{cite book | author=Florescu, Ionut| title=Probability and Stochastic Processes| publisher=Wiley| year=2014 | isbn=978-0-470-62455-5}}</ref>{{rp|p. 29}}<ref name=Gallager/>{{rp|p. 10}}

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<math>A \cap B \neq \emptyset</math> indicates that two independent events <math>A</math> and <math>B</math> have common elements in their [[sample space]] so that they are not [[Mutual exclusivity|mutually exclusive]] (mutually exclusive iff <math>A \cap B = \emptyset</math>). Why this defines independence is made clear by rewriting with [[Conditional probability|conditional probabilities]] <math>P(A \mid B) = \frac{P(A \cap B)}{P(B)}</math> as the probability at which the event <math>A</math> occurs provided that the event <math>B</math> has or is assumed to have occurred:

:<math>\mathrm{P}(A \cap B) = \mathrm{P}(A)\mathrm{P}(B) \iff \mathrm{P}(A\mid B) = \frac{\mathrm{P}(A \cap B)}{\mathrm{P}(B)} = \mathrm{P}(A).</math>

and similarly

:<math>\mathrm{P}(A \cap B) = \mathrm{P}(A)\mathrm{P}(B) \iff\mathrm{P}(B\mid A) = \frac{\mathrm{P}(A \cap B)}{\mathrm{P}(A)} = \mathrm{P}(B).</math>

Thus, the occurrence of <math>B</math> does not affect the probability of <math>A</math>, and vice versa. In other words, <math>A</math> and <math>B</math> are independent of each other. Although the derived expressions may seem more intuitive, they are not the preferred definition, as the conditional probabilities may be undefined if <math>\mathrm{P}(A)</math> or <math>\mathrm{P}(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>.