Conditional probability: Difference between revisions

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{{Short description|Probability of an event occurring, given that another event has already occurred}}

{{Probability fundamentals}}

In [[probability theory]], '''conditional probability''' is a measure of the [[probability]] of an [[Event (probability theory)|event]] occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred.<ref name="Allan Gut 2013">{{cite book |last=Gut |first=Allan |title=Probability: A Graduate Course |year=2013 |publisher=Springer |___location=New York, NY |isbn=978-1-4614-4707-8 |edition=Second }}</ref> This particular method relies on event B occurring with some sort of relationship with another event A. In this event, the event B can be analyzed by a conditionallyconditional probability with respect to A. If the event of interest is {{mvar|A}} and the event {{mvar|B}} is known or assumed to have occurred, "the conditional probability of {{mvar|A}} given {{mvar|B}}", or "the probability of {{mvar|A}} under the condition {{mvar|B}}", is usually written as {{math|P(''A''{{!}}''B'')}}<ref name=":0">{{Cite web|title=Conditional Probability|url=https://www.mathsisfun.com/data/probability-events-conditional.html|access-date=2020-09-11|website=www.mathsisfun.com}}</ref> or occasionally {{math|P{{sub|''B''}}(''A'')}}. This can also be understood as the fraction of probability B that intersects with A: <math>P(A \mid B) = \frac{P(A \cap B)}{P(B)}</math>.<ref>{{Cite journal|last1=Dekking|first1=Frederik Michel|last2=Kraaikamp|first2=Cornelis|last3=Lopuhaä|first3=Hendrik Paul|last4=Meester|first4=Ludolf Erwin|date=2005|title=A Modern Introduction to Probability and Statistics|url=https://doi.org/10.1007/1-84628-168-7|journal=Springer Texts in Statistics|language=en-gb|pages=26|doi=10.1007/1-84628-168-7|isbn=978-1-85233-896-1 |issn=1431-875X}}</ref>

For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person is sick, then they are much more likely to be coughing. For example, the conditional probability that someone unwell (sick) is coughing might be 75%, in which case we would have that {{math|P(Cough)}} = 5% and {{math|P(Cough{{!}}Sick)}} = 75 %. Although there is a relationship between {{mvar|A}} and {{mvar|B}} in this example, such a relationship or dependence between {{mvar|A}} and {{mvar|B}} is not necessary, nor do they have to occur simultaneously.