# Lec05 - IOE/Stat 265, Fall 2009 Lecture #5: Bayes Rules!...

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1 IOE/Stat 265, Fall 2009 IOE/Stat 265, Fall 2009 Lecture #5: Lecture #5: Bayes Bayes Rules! Rules! 2 Independence Two events A and B are independent if P(A|B) = P(A) This implies that the occurrence of event B doesn’t have an effect on the occurrence of event A If P(A|B) ± P(A) , then A and B are dependent . So, if two events are mutually exclusive , are they independent ? 4 Independence - Definition 5 Independent Events If and only if (iff) A and B are independent If you have a set of mutually independent events (A 1 .. A n ), then ! () ( ) * ( ) PA B PA PB ! ! " 2 1 ( ) n ni i PA A A PA

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6 Example 1: HDTV Process Control High Definition TVs display 1920 x 1080 pixels. The probability of a defective pixel is “p” Assuming pixels function independently, what is the probability no pixels fail to operate? 9 Independence and Reliability: Series Systems ! Reliability – probability that a system does not fail given a period of time under certain operating conditions. ! Series Reliability (one unit fails, system fails) R 1 R 2 i = units in series ! ! ! " 1 12 () * n si i s RR t R 10 Independence and Reliability: Parallel Systems ! Parallel System Reliability (if either unit survives, system survives) R 1 R 2 ## " n i=1 = 1 (1 ) 11 Example 2: Twin Engine Airplane Twin Engine Plane Example: If both engines are independent and if P(Engine Survives) = 0.99 What is the probability that both will survive? What is the probability that at least one will survive? What is the purpose of adding redundancy to a system (i.e., multiple components performing the same function)? Another Example: alarm clock & wake-up service
15 Example 3: Collections of Independent Systems Suppose you have a complex systems of independent components. To operate, component 3 must survive and either component 1 or 2.

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## This note was uploaded on 03/17/2010 for the course IOE 265 taught by Professor Garyherrin during the Fall '09 term at University of Michigan-Dearborn.

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Lec05 - IOE/Stat 265, Fall 2009 Lecture #5: Bayes Rules!...

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