prob_Bertsekas-Ch1

# prob_Bertsekas-Ch1 - 1 Sample Space and Probability...

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1 Sample Space and Probability Contents 1.1. Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 3 1.2. Probabilistic Models . . . . . . . . . . . . . . . . . . . . p. 6 1.3. Conditional Probability . . . . . . . . . . . . . . . . . p. 18 1.4. Total Probability Theorem and Bayes’ Rule . . . . . . . . p. 28 1.5. Independence . . . . . . . . . . . . . . . . . . . . . . p. 34 1.6. Counting . . . . . . . . . . . . . . . . . . . . . . . p. 44 1.7. Summary and Discussion . . . . . . . . . . . . . . . . p. 51 Problems . . . . . . . . . . . . . . . . . . . . . . . . p. 53 1

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2 Sample Space and Probability Chap. 1 “Probability” is a very useful concept, but can be interpreted in a number of ways. As an illustration, consider the following. A patient is admitted to the hospital and a potentially life-saving drug is administered. The following dialog takes place between the nurse and a concerned relative. RELATIVE : Nurse, what is the probability that the drug will work? NURSE : I hope it works, we’ll know tomorrow. RELATIVE : Yes, but what is the probability that it will? NURSE : Each case is different, we have to wait. RELATIVE : But let’s see, out of a hundred patients that are treated under similar conditions, how many times would you expect it to work? NURSE (somewhat annoyed): I told you, every person is different, for some it works, for some it doesn’t. RELATIVE (insisting): Then tell me, if you had to bet whether it will work or not, which side of the bet would you take? NURSE (cheering up for a moment): I’d bet it will work. RELATIVE (somewhat relieved): OK, now, would you be willing to lose two dollars if it doesn’t work, and gain one dollar if it does? NURSE (exasperated): What a sick thought! You are wasting my time! In this conversation, the relative attempts to use the concept of probability to discuss an uncertain situation. The nurse’s initial response indicates that the meaning of “probability” is not uniformly shared or understood, and the relative tries to make it more concrete. The first approach is to define probability in terms of frequency of occurrence , as a percentage of successes in a moderately large number of similar situations. Such an interpretation is often natural. For example, when we say that a perfectly manufactured coin lands on heads “with probability 50%,” we typically mean “roughly half of the time.” But the nurse may not be entirely wrong in refusing to discuss in such terms. What if this was an experimental drug that was administered for the very first time in this hospital or in the nurse’s experience? While there are many situations involving uncertainty in which the fre- quency interpretation is appropriate, there are other situations in which it is not. Consider, for example, a scholar who asserts that the Iliad and the Odyssey were composed by the same person, with probability 90%. Such an assertion conveys some information, but not in terms of frequencies, since the subject is a one-time event. Rather, it is an expression of the scholar’s subjective be- lief . One might think that subjective beliefs are not interesting, at least from a mathematical or scientific point of view. On the other hand, people often have to make choices in the presence of uncertainty, and a systematic way of making use of their beliefs is a prerequisite for successful, or at least consistent, decision making.
Sec. 1.1 Sets 3

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