Editing Mutual exclusivity (section)

==Probability==
In [[probability theory]], events ''E''<sub>1</sub>, ''E''<sub>2</sub>, ..., ''E''<sub>''n''</sub> are said to be mutually exclusive if the occurrence of any one of them implies the non-occurrence of the remaining ''n''&nbsp;−&nbsp;1 events. Therefore, two mutually exclusive events cannot both occur. Formally said, <math> X </math> is a set of mutually exclusive events [[if and only if]] given any <math> E_i, E_j \in X </math>, if <math> E_i \neq E_j </math> then <math> E_i \cap E_j = \varnothing </math>. As a consequence, mutually exclusive events have the property: <math> P(A \cap B) = 0 </math>.<ref>[http://www.intmath.com/Counting-probability/9_Mutually-exclusive-events.php intmath.com]; Mutually Exclusive Events. Interactive Mathematics. December 28, 2008.</ref>

For example, in a [[standard 52-card deck]] with two colors it is impossible to draw a card that is both red and a club because clubs are always black. If just one card is drawn from the deck, either a red card (heart or diamond) or a black card (club or spade) will be drawn. When ''A'' and ''B'' are mutually exclusive, {{nowrap|1=P(''A'' ∪ ''B'') = P(''A'') + P(''B'')}}.<ref name="rules">[http://people.richland.edu/james/lecture/m170/ch05-rul.html Stats: Probability Rules.]</ref> To find the probability of drawing a red card or a club, for example, add together the probability of drawing a red card and the probability of drawing a club. In a standard 52-card deck, there are twenty-six red cards and thirteen clubs: 26/52 + 13/52 = 39/52 or 3/4.

One would have to draw at least two cards in order to draw both a red card and a club. The probability of doing so in two draws depends on whether the first card drawn was replaced before the second drawing since without replacement there is one fewer card after the first card was drawn. The probabilities of the individual events (red, and club) are multiplied rather than added. The probability of drawing a red and a club in two drawings without replacement is then {{nowrap|1=26/52 × 13/51 × 2 = 676/2652}}, or 13/51. With replacement, the probability would be {{nowrap|1=26/52 × 13/52 × 2 = 676/2704}}, or 13/52.

In probability theory, the word ''or'' allows for the possibility of both events happening. The probability of one or both events occurring is denoted P(''A'' ∪ ''B'') and in general, it equals P(''A'') + P(''B'') – P(''A'' ∩ ''B'').<ref name="rules" /> Therefore, in the case of drawing a red card or a king, drawing any of a red king, a red non-king, or a black king is considered a success. In a standard 52-card deck, there are twenty-six red cards and four kings, two of which are red, so the probability of drawing a red or a king is 26/52 + 4/52 – 2/52 = 28/52.

Events are [[collectively exhaustive]] if all the possibilities for outcomes are exhausted by those possible events, so at least one of those outcomes must occur. The probability that at least one of the events will occur is equal to one.<ref name="events">[https://web.archive.org/web/20110720051423/http://www.cs.stedwards.edu/chem/Chemistry/CHEM4341/BayesPrimer2.pdf Scott Bierman. A Probability Primer. Carleton College. Pages 3-4.]</ref> For example, there are theoretically only two possibilities for flipping a coin. Flipping a head and flipping a tail are collectively exhaustive events, and there is a probability of one of flipping either a head or a tail. Events can be both mutually exclusive and collectively exhaustive.<ref name="events" /> In the case of flipping a coin, flipping a head and flipping a tail are also mutually exclusive events. Both outcomes cannot occur for a single trial (i.e., when a coin is flipped only once). The probability of flipping a head and the probability of flipping a tail can be added to yield a probability of 1: 1/2 + 1/2 =1.<ref>{{Cite web |url=http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/NonMutually-Exclusive-Outcomes.topicArticleId-25951,articleId-25914.html |title=Non-Mutually Exclusive Outcomes. CliffsNotes. |access-date=2009-07-10 |archive-url=https://web.archive.org/web/20090528210738/http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/NonMutually-Exclusive-Outcomes.topicArticleId-25951,articleId-25914.html |archive-date=2009-05-28 |url-status=dead }}</ref>