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A student at a certain university will pass the oral Ph D qualifying examination if at least two of the three examiners pass her or him Past experience shows that a 15% of the students who take

A student at a certain university will pass the oral Ph.D. qualifying examination if at least two of the three examiners pass her or him. Past experience shows that (a) 15% of the students who take the qualifying exam are not prepared, and (b) each examiner will independently pass 85% of the prepared and 20% of the unprepared students. Kevin took his Ph.D. qualifying exam with Professors Smith, Brown, and Rose. What is the probability that Professor Rose has passed Kevin if we know that neither Professor Brown nor Professor Smith has passed him? Let S, B, and R be the respective events that Professors Smith, Brown, and Rose have passed Kevin. Are these three events independent? Are they conditionally independent given that Kevin is prepared? (See Exercises 43 and 44, Section 3.5.)

Exercise 43

Hemophilia is a hereditary disease. If a mother has it, then with probability 1/2, any of her sons independently will inherit it. Otherwise, none of the sons becomes hemophilic. Julie is the mother of two sons, and from her family’s medical history it is known that, with the probability 1/4, she is hemophilic. What is the probability that

(a) her first son is hemophilic;

(b) her second son is hemophilic;

(c) none of her sons are hemophilic? Hint: Let H, H1, and H2 denote the events that the mother, the first son, and the second son are hemophilic, respectively. Note that H1 and H2 are conditionally independent given H. That is, if we are given that the mother is hemophilic, knowledge about one son being hemophilic does not change the chance of the other son being hemophilic. However, H1 and H2 are not independent. This is because if we know that one son is hemophilic, the mother is hemophilic and therefore with probability 1/2 the other son is also hemophilic.

Exercise 44

(Laplace’s Law of Succession) Suppose that n + 1 urns are numbered 0 through n, and the ith urn contains i red and n − i white balls, 0 ≤ i ≤ n. An urn is selected at random, and then the balls it contains are removed one by one, a random, and with replacement. If the first m balls are all red, what is the probability that the (m + 1)st ball is also red? Hint: Let Ui be the event that the ith urn is selected, Rm the event that the first m balls drawn are all red, and R the event that the (m + 1)st ball drawn is red. Note that R and Rm are conditionally independent given Ui; that is, given that the ith urn is selected, R and Rm are independent. Hence

Laplace designed and solved this problem for philosophical reasons. He used it to argue that the sun will rise tomorrow with probability (m+1)/(m+2) if we know that it has risen in the m preceding days. Therefore, according to this, as time passes by, the probability that the sun rises again becomes higher, and the event of the sun rising becomes more and more certain day after day. It is clear that, to argue this way, we should accept the problematic assumption that the phenomenon of the sun rising is “random,” and the model of this problem is identical to its random behavior.

Aug 03 2020 View more View Less

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