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l18_prob_cond - 6.042/18.062J Mathematics for Computer...

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6.042/18.062J Mathematics for Computer Science April 21, 2005 Srini Devadas and Eric Lehman Lecture Notes Conditional Probability 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, we’re picking a random point in the big ellipse shown below and asking how likely that point is to fall into region A or B : A B set of all people in the world set of people who live in Cambridge set of MIT students 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 is the probability that a person is an MIT student, given that the person lives in Cambridge? This should be much greater— but what it is exactly? What we’re 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? • What is the probability that two rolled dice sum to 10, given that both are odd? • What is the probability that I’ll get four-of-a-kind in Texas No Limit Hold ’Em Poker, given that I’m initially dealt two queens? There is a special notation for conditional probabilities. In general, Pr ( A | B ) denotes the probability of event A , given that event B happens. So, in our example, Pr ( A | B ) is the probability that a random person is an MIT student, given that he or she is a Cam- bridge resident.
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2 Conditional Probability How do we compute Pr ( A | 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 | 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 (darkly shaded) divided by the probability that the person is in set B (lightly shaded). This motivates the definition of conditional probability: Pr ( A | B ) = Pr ( A B ) Pr ( B ) If Pr ( B ) = 0 , then the conditional probability Pr ( A | B ) is undefined. Probability is generally counterintuitive, but conditional probability is the worst! Con- ditioning 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 trouble— provided you rely on formal reasoning and not intuition.
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