Prob_Primer - Dr. Maddah ENMG 500 Engg Management I...

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1 Dr. Maddah ENMG 500 Eng’g Management I 01/18/09 Probability Primer 1 Sample space and Events ¾ Suppose that an experiment with an uncertain outcome is performed (e.g., rolling a die). ¾ While the outcome of the experiment is not known in advance, the set of all possible outcomes is known. This set is the sample space, Ω . ¾ For example, when rolling a die Ω = {1, 2, 3, 4, 5, 6}. When tossing a coin, Ω = {H, T}. When measuring life time of a machine (years), Ω = {1, 2, 3, …}. ¾ A subset E Ω is known as an event. ¾ For example, when rolling a die, E = {1} is the event that one appears. The subset E = {1, 3, 5} is the event that an odd number appears. Probability of an event ¾ If an experiment is repeated for a number of times which is large enough, the fraction of time that event E occurs is the probability that event E occurs, P { E }. ¾ E.g., when rolling a fair die, P {1} = 1/6, and P {1, 3, 5} = 3/6 = 1/2. When tossing a fair coin, P {H} = P {T} = 1/2. 1 Compiled from S. M. Ross, Introduction to Probability Models , 6 th Edition, Academic Press, 1997.
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2 Axioms of probability (1) For E Ω , 0 P { E } 1; (2) P { Ω} = 1; (3) For events E 1 , E 2 , …, E i , …, with E i Ω, E i E j = , for all i and j , 1 1 {} ii i i P EP E = = ⎧⎫ = ⎨⎬ ⎩⎭ . Implications ¾ The axioms of probability imply the following results: o For E and F Ω , P { E “or” F } = P { E F } = P { E } + P { F } P { E F } ; 2 o If E and F are mutually exclusive (i.e., E F = ), then P { E F } = P { E } + P { F }; o For E Ω , let E c be the complement of E (i.e., E E c = Ω ), P { E c } = 1 P { E }; o P { } = 0. Conditional probability ¾ The probability that event E occurs given that event F has already occurred is {|} PE F PF = .
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This note was uploaded on 12/23/2010 for the course ENMG 500 taught by Professor Drbacelmaddah during the Fall '09 term at American University of Beirut.

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Prob_Primer - Dr. Maddah ENMG 500 Engg Management I...

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