lec0118 - IEOR 4106: Introduction to Operations Research:...

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IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2011, Professor Whitt Class Lecture Notes: Tuesday, January 18 Conditional Probability We discussed several problems in Chapter 1 of Ross. I emphasized several key points: 1. Probability theory is a branch of mathematics , so it is important to pay attention to deFnitions and axioms (see the beginning of Section 1.3 in the Ross textbook). You need to recall some elementary set theory . We use braces to denote a set, as appear in the deFnition of the events S and E in Problem 1.18 below. Just to focus on one pedantic (but useful) detail, note that x , { x } and {{ x }} are di±erent objects: x is an element of the set { x } , while { x } is an element of the set {{ x }} ; x is not an element of the set {{ x }} . A set containing the element x is not the same as the element x itself. It may help to Google “set theory.” That is, look at Wikipedia, PlanetMath or Wolfram’s Mathworld. One or two pages of reading should su²ce. ³ormally, a probability measure assigns probabilities to subsets of the sample space; those subsets are called events. See Sections 1.1-1.3 of Ross. 2. We are focusing on problem solving . ³or that purpose, a good general strategy is divide and conquer : break the problem into smaller pieces that are easier to analyze. Skipping steps can cause errors. 3. It is helpful to draw pictures . In particular, Chapter 1 emphasizes that a key idea overall is to remember and apply the deFnition of conditional probability : P ( A | B ) P ( AB ) P ( B ) , where AB A B denotes the intersection of the events (sets) A and B . The following are exercises at the end of Chapter 1 in Ross. 1.18 (a) A family has two children. What is the probability that both are girls, given that at least one is a girl? (Assume that each child is equally likely to be a boy or a girl.) (b) Does the answer change if we rephrase the question: What is the probability that both
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lec0118 - IEOR 4106: Introduction to Operations Research:...

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