Solution Manual For Probability, Statistics, and Random Processes for Engineers, 4th Edition

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1Solutions to Chapter 11. The intent of this rather vague problem is to get you to compare the two notions, probabilityas intuition and relative frequency theory.There are many possible answers to how tomake the statement "Ralph is probably guilty of theft" have a numerical value in the relativefrequency theory.First step is to define a repeatable experiment along with its outcomes.The favorable outcome in this case would be ’guilty.’Repeating this experiment a largenumber of times would then give the desired probability in a relative frequency sense.Wethus see that it may entail a lot of work to attach an objective numerical value to such asubjective statement, if in fact it can be done at all.One possible approach would be to look through courthouse statistics for cases similar toRalph’s, similar both in terms of the case itself and the defendant.If we found a sufficientlylarge number of these cases, ten at least, we could then form the probability=, whereis the number of favorable (guilty) verdicts,andis the total number of found cases.Here we effectively assume that the judge and jury are omniscient.Another possibility is tofind a large number of people with personalities and backgroundssimilar to Ralph’s, and to expose them to a very similar situation in which theft is possible.The fraction of these people that then steal in relation to the total number of people, wouldthen give an objective meaning to the phrase "Ralph is probably guilty of theft."2. Note that→3, but36→, i.e.,implies3but not the other way around. Thus if weturn over card 2 andfind a3. So what? It was never stated that a3→. Likewise, withcard 3.On the other hand, if we turn over card 4 andfind a, then the rule is violated.Hence, we must turn over card 4 and card 1, of course.3. First step here is to decide which kind of probability to use.Since no probabilities areexplicitly given, it is reasonable to assume that all numbers are equally likely.Effectivelywe assume that the wheel is “fair."This then allows us to use the classical theory alongwith the axiomatic theory to solve this problem. Now we mustfind the corresponding prob-ability model.We are told in the problem statement that the experiment is “spinning thewheel." We identify the pointed-to numbers as the outcomes.The sample space is thusΩ={123456789}The total number of outcomes is then 9. The probability of eachelemental event{}is then taken as[{}],= 19, as in the classical theory. We are alsotold in the problem statement that the contestant wins if an even number shows. The set ofeven numbers inΩis{2468}We can write this event as a disjoint union of four singleton(atomic) events{2468}={2}∪{4}∪{6}∪{8}Now we can apply axiom 3 of probability to write[{2468}]=[{2}] +[{4}] +[{6}] +[{8}]=19 + 19 + 19 + 19=49We have seen that some ’reasonable’ assumptions are necessary to transform the given wordproblem into something that exactly corresponds to a probability model.It turns out thatthis is a general problem for such word problems, i.e. problems given in natural English.

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