Conditional probability: Difference between revisions

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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) is already known to have 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 A occurring with some sort of relationship with another event B. In this situation, the event A can be analyzed by a conditional probability with respect to B. 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, or the ratio of the probabilities of both events happening to the "given" one happening (how many times A occurs rather than not assuming B has occurred): <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.

{{math|P(''A''{{!}}''B'')}} may or may not be equal to {{math|P(''A'')}}, i.e., the '''unconditional probability''' or '''absolute probability''' of {{mvar|A}}. If {{math|1=P(''A''{{!}}''B'') = P(''A'')}}, then events {{mvar|A}} and {{mvar|B}} are said to be [[Independence (probability theory)#Two events|''independent'']]: in such a case, knowledge about either event does not alter the likelihood of each other. {{math|P(''A''{{!}}''B'')}} (the conditional probability of {{mvar|A}} given {{mvar|B}}) typically differs from {{math|P(''B''{{!}}''A'')}}. For example, if a person has [[dengue fever]], the person might have a 90% chance of being tested as positive for the disease. In this case, what is being measured is that if event {{mvar|B}} (''having dengue'') has occurred, the probability of {{mvar|A}} (''tested as positive'') given that {{mvar|B}} occurred is 90%, simply writing {{math|P(''A''{{!}}''B'')}} = 90%. Alternatively, if a person is tested as positive for dengue fever, they may have only a 15% chance of actually having this rare disease due to high [[false positive]] rates. In this case, the probability of the event {{mvar|B}} (''having dengue'') given that the event {{mvar|A}} (''testing positive'') has occurred is 15% or {{math|P(''B''{{!}}''A'')}} = 15%. It should be apparent now that falsely equating the two probabilities can lead to various errors of reasoning, which is commonly seen through [[base rate fallacy|base rate fallacies]].