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Unformatted text preview: 15 mcs-ftl 2010/9/8 0:40 page 417 #423 Conditional Probability 15.1 Definition Suppose that we pick a random person in the world. Everyone has an equal chance of being selected. Let A be the event that the person is an MIT student, and let B be the event that the person lives in Cambridge. What are the probabilities of these events? Intuitively, were picking a random point in the big ellipse shown in Figure 15.1 and asking how likely that point is to fall into region A or B . set of all people in the world set of people who live in Cambridge set of MIT students B A Figure 15.1 Selecting a random person. A is the event that the person is an MIT student. B is the even that the person lives in Cambridge. The vast majority of people in the world neither live in Cambridge nor are MIT students, so events A and B both have low probability. But what about the prob- ability that a person is an MIT student, given that the person lives in Cambridge? This should be much greaterbut what is it exactly? What were asking for is called a conditional probability ; that is, the probability that one event happens, given that some other event definitely happens. Questions about conditional probabilities come up all the time: What is the probability that it will rain this afternoon, given that it is cloudy this morning? 418 mcs-ftl 2010/9/8 0:40 page 418 #424 Chapter 15 Conditional Probability What is the probability that two rolled dice sum to 10, given that both are odd? What is the probability that Ill get four-of-a-kind in Texas No Limit Hold Em Poker, given that Im initially dealt two queens? There is a special notation for conditional probabilities. In general, Pr A j B denotes the probability of event A , given that event B happens. So, in our example, Pr A j B is the probability that a random person is an MIT student, given that he or she is a Cambridge resident. How do we compute Pr A j B ? Since we are given that the person lives in Cambridge, we can forget about everyone in the world who does not. Thus, all outcomes outside event B are irrelevant. So, intuitively, Pr A j B should be the fraction of Cambridge residents that are also MIT students; that is, the answer should be the probability that the person is in set A \ B (the darkly shaded region in Figure 15.1 ) divided by the probability that the person is in set B (the lightly shaded region). This motivates the definition of conditional probability: Definition 15.1.1. Pr A \ B Pr B A j WWD Pr B If Pr B D , then the conditional probability Pr A j B is undefined. Pure probability is often counterintuitive, but conditional probability is even worse! Conditioning can subtly alter probabilities and produce unexpected results in randomized algorithms and computer systems as well as in betting games. Yet, the mathematical definition of conditional probability given above is very simple and should give you no troubleprovided that you rely on formal reasoning and...
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- Fall '10
- Computer Science