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hw1 - cancer doctors often perform a test that measures the...

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Problem Set 1 ORIE360/560, Fall 2004 Yoonjung Lee Posted on September 3, 2004 Due on September 10, 2004 1. (Problem 17, Chapter 3) Show that n r = n - 1 r - 1 + n - 1 r . For a combinatorial argument, consider a set of n items and fix atten- tion on one of these items. How many different sets of size r contain this item, and how many do not? 2. Prove the following statement: If A , B , and C are mutually indepen- dent events, then A c and ( B C c ) are independent. 3. Let A and B be two events. (a) If the events A and B are mutually exclusive, are A and B always independent? If the answer is no, can they ever be independent? Explain. (b) If A B , can A and B be ever independent events? Explain. 4. A single card is drawn at random from each of six well-shuffled decks of playing cards. Let A be the event that all six cards are different. (a) Find P ( A ). (b) Find the probability that at least two of the drawn cards match. 1
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5. (Problem 34, Chapter 3) Prostate cancer is the most common type of cancer found in males. As an indicator of whether a male has prostate
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Unformatted text preview: cancer, doctors often perform a test that measures the level of the PSA protein (prostate specific antigen) that is produced only by the prostate gland. Although higher PSA levels are indicative of cancer, the rest is notoriously unreliable. Indeed, the probability that a noncancerous man will have an elevated PSA level is approximately . 135, with this probability increasing to approximately . 268 if the man does have can-cer. If, based on other factors, a physician is 70 percent certain that a male has prostate cancer, what is the conditional probability that he has the cancer given that (a) the test indicates an elevated PSA level; (b) the test does not indicate an elevated PSA level? Repeat the preceding, this time assuming that the physician initially believes there is a 30 percent chance the man has prostate cancer. 2...
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