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| CATEGORIES ABOUT CONDITIONAL PROBABILITY | |
| probability theory | |
| logical fallacies | |
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Conditional probability is the Probability of some Event ''A'', given the occurrence of some other event ''B''. | ||
| :<math>P(AB) \ | \ P(A)</math> |
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| :<math>P(BA) \ | \ P(B)</math> |
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| If <math>B</math> Is An Event And <math>P(B) > 0</math>, Then The Function <math>Q</math> Defined By <math>Q(A) | P(AB)</math> for all events <math>A</math> is a Probability Measure |
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| The Conditional Probability | "http://wwwinformationdelightinfo/information/entry/fallacy" class="copylinks">Fallacy is the assumption that ''P''(''A''''B'') is approximately equal to ''P''(''B''''A'') The mathematician John Allen Paulos discusses this in his book Innumeracy , where he points out that it is a mistake often made even by doctors, lawyers, and other highly educated non- Statistician s It can be overcome by describing the data in actual numbers rather than probabilities |
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| The Relation Between ''P''(''A''''B'') And ''P''(''B''''A'') Is Given By | "http://wwwinformationdelightinfo/information/entry/Bayes_Theorem" class="copylinks">Bayes Theorem : |
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| :<math>P( Ext{positive} Ext{well}) | 1%</math>, and <math>P( ext{negative} ext{well})=99%</math> |
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| :<math>P( Ext{negative} Ext{disease}) | 1%</math> and <math>P( ext{positive} ext{disease})=99%</math> |
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| :<math>P( Ext{well}\cap Ext{negative}) | P( ext{well}) imes P( ext{negative} ext{well})=99% imes99%=9801%</math> is the fraction of the whole group being well and testing negative |
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| :<math>P( Ext{disease}\cap Ext{positive}) | P( ext{disease}) imes P( ext{positive} ext{disease})=1% imes99%=099%</math> is the fraction of the whole group being ill and testing positive |
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| :<math>P( Ext{well}\cap Ext{positive}) | P( ext{well}) imes P( ext{positive} ext{well})=99% imes1%=099%</math> is the fraction of the whole group having false positive results |
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| :<math>P( Ext{disease}\cap Ext{negative}) | P( ext{disease}) imes P( ext{negative} ext{disease})=1% imes1%=001%</math> is the fraction of the whole group having false negative results |
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| :<math>P( Ext{disease} Ext{positive}) | rac{P( ext{disease}\cap ext{positive})}{P( ext{positive})}= rac{099%}{198%}=50%</math> is the probability that you actually have the disease if you tested positive |
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| In This Example, It Should Be Easy To Relate To The Difference Between P(positivedisease) | 99% and P(diseasepositive)=50%: The first is the conditional probability that you test positive if you have the disease the second is the conditional probability that you have the disease if you test positive With the numbers chosen here, the last result is likely to be deemed unacceptable: Half the people testing positive are actually false positives |
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