204 5.6 bayes thoerem

204 5.6 bayes thoerem - 5.6 CONDITIONAL PROBABILTY and...

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5.6 CONDITIONAL PROBABILTY and BAYES’ THEOREM Bayes’ theorem is used when we know the conditional of A given B but want to calculate the conditionalof B given A. For example, assume we have data that show that 10% of all deaths are due to lung cancer and 40% of lung cancer deaths (L) are for people that smoked(S). We also have data that indicate 20% of the population smokes. What we don’t know is the probability of dying of lung cancer given that an individual smokes. i.e. We know P(S|L) but would like to know P(L|S). For this we use Bayes’ theorem. From our study of conditional probability we know: (1a) P(B | A) = P(AB) / P(A) and (1b) P(A|B) = P(AB)/P(B) Multiplying these equations by their denominators we get: (2a) P(A) P(B|A) = P(AB) and (2b) P(B) P(A|B) = P(AB) Consider the two ways that A can occur; It can occur with B and it can occur without B. ( i.e. AB or AB c ) Note that AB and AB c are mutually exclusive thus: P(A) = P(AB or AB c ) = P(AB) + P(AB c ) substituting this result into (1) gives
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This note was uploaded on 12/05/2011 for the course MATH 2040 taught by Professor Raysievers during the Fall '10 term at Utah Valley University.

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204 5.6 bayes thoerem - 5.6 CONDITIONAL PROBABILTY and...

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