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Unformatted text preview: Lecture 10 Introduction to Probability Topics Events, sample space, random variables Examples Probability distribution function Conditional probabilities Exponential distribution Poisson distribution There is $1,000,000 behind one door & $0.00 behind the other two. You reserve a door (say #1) but it remains closed . Then, Monte opens one of the other two doors (say #2). The door Monte opens will be empty ! (Monte wants to keep the money.) #1 #2 #3 The Monte Hall Problem 1. Stay with the original door we chose? 2. Switch to the other unopened door? Whats the best strategy? We are interested in probability as it relates to helping us make good or optimal decisions. Coins & Drawers One drawer has 2 gold coins One drawer has 1 gold & 1 silver coin One drawer has 2 silver coins You select a drawer at random and then randomly select one coin It turns out that your coin is gold What is the probability that the other coin is gold ? Another Example Probability Overview The sample space , S = set of all possible outcomes of an experiment Events are subsets of the sample space. An event occurs if any of its elements occur when the experiment is run. Two events A & B are mutually exclusive if their intersection is null . Elements of S are called realizations outcomes , sample points , or scenarios The choice of the sample space depends on the question that you are trying to answer. Two possible sample spaces are: S 1 = { (1,1), (1,2), (1,3), , (6,4), (6,5), (6,6) } S 2 = { 2, 3, 4, . . . , 11, 12 } (sum of the two values) Examples of Events : A 1 = the sum of the face values is 3 Under S 1 : A 1 = { (1,2), (2,1) } ; Under S 2 : A 1 = { 3 } A 2 = one die has value 3 & the other has value 1 Roll Two Dice Under S 1 : A 2 = { (1,3), (3,1) } ; Under S 2 : not an event Mathematical : A probability measure P is defined on the set of all events satisfying the following: (1) P(A) 0 2200 A S (2) P(S) = P(A 1 A 2 ) = 1 (3) Mutually exclusive events A i imply that P(A 1 A 2 ) = P(A 1 ) + P(A 2 ) + (4) P(A) = 1  P(A) where A = complement of A (5) P(A B) = P(A) + P(B)  P(A B) (6) If A & B are independent , then P(A B) = P(A)P(B) _ _ Probability Intuitive : The proportion of time that an event occurs when the same experiment is repeated (7) If A & B are...
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This note was uploaded on 12/19/2011 for the course M E 366l taught by Professor Staff during the Spring '08 term at University of Texas at Austin.
 Spring '08
 Staff

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