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1. Freddy is a scientist that is testing a vaccine for the coronavirus. The vaccine is equally likely to work or not. Freddy believes in

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1. Freddy is a scientist that is testing a vaccine for the coronavirus. The vaccine is equally likely to work or not. Freddy believes in the law of small numbers; his urn contains N = 5 balls (some that say that the vaccine works, and some that say it doesn't). He plans on conducting a number of independent experiments, each of which will test whether the vaccine is effective against coronavirus. If the vaccine works, the results from the experiment will confirm it with 80% probability, and if the vaccine does not work, the results from the experiment will reject it with 80% probability. Freddy will stop running experiments (and exploring new vaccine options) once he is convinced that the vaccine works. (a) Suppose initially that Freddy believes the vaccine works for sure. Before he runs any tests, what does he think is the probability that the first test will show that the vaccine works? If the first test finds the vaccine works, what will Freddy believe is the probability that the second test will show that the vaccine works? (b) How does the second probability in part (a) compare to the first one? What phenomenon does this reflect? Explain. (c) Now suppose that Freddy is unsure whether the vaccine works or not. The vaccine can either work, wherein 80% of the balls in Freddy's urn confirm the result, or does not work wherein 80% of the balls in Freddy's urn reject the result. Freddy begins to test his theory. Assuming that he runs two experiments, and both find that the vaccine does not work, what does Freddy believe is the likelihood that the vaccine works? Will he run any more tests? (d) What phenomenon does this reflect? Explain. (e) Use Bayes Rule to compute the likelihood that the vaccine does work, given two results showing it does not work, Pr(vaccine works | two non-works results)

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