ioe265f11-Lec05 - Lec05 Bayes Theorem and Independence...

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Lec05 – Bayes Theorem and Independence IOE 265 F11 1 1 Bayes Theorem Independence 2 Topics Conditional Probability Multiplication Rule Law of Total Probability Bayes Theorem Independence
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Lec05 – Bayes Theorem and Independence IOE 265 F11 2 3 Conditional Probability Conditional probability is used when the outcome of an event may change given the outcome of a related event. P(A|B) = Prob of A given B 0 ) ( ) ( ) ( ) | ( B P for B P B A P B A P B A Given B, what’s the probability of A? In a conditional probability problem, the sample space is “reduced” to the “space” of the given outcome (e.g. if given B, we now just care about the probability of A occurring “inside” of B) 4 Conditional Probability -- Interpretation How should we interpret the P(B | A)? If all outcomes of an experiment are equally likely and there are n total outcomes, then: P(A) = (number of A outcomes) / n P(A B) = (number of outcomes in A and B)/n So, P(B|A) = outcomes of A and B / outcomes of A So, P(B | A) represents the relative frequency of event B among the trials that produce an outcome in event A. 0 ) ( ) ( / ) ( ) | ( A P for A P B A P A B P U
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IOE 265 F11 3 5 Conditional Probability Relationships The intersection of two events may be re-written from the above using the “multiplication rule”. Multiplication rule is useful for determining probability of an event that depends on other events. So, 0 ) ( ) ( / ) ( ) | ( A P for A P B A P A B P 0 ) ( ) ( / ) ( ) | ( B P for B P B A P B A P Conditional Probability Relationships: ) ( * ) | ( ) ( * ) | ( ) ( A P A B P B P B A P B A P 6 Multiplication Rule P(A B) = P(B) * P(A|B) Or P(A)* P(B|A) Examples: Car Crash Injuries: Event A – Getting injured in a crash (injuries: hospital visits or fatalities) Event B – US resident involved in a car crash; P(B) = 0.01; P(A | B) = 0.30; -- Of those US residents involved in crashes, 30% get injured P (A B) = What is the probability a US resident will be in a car crash and get injured?
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ioe265f11-Lec05 - Lec05 Bayes Theorem and Independence...

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