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Next: Conditional Independence [2 P] Up: MLA_Exercises_2007 Previous: Exercises

Bayes' Rule [4+1* P]

a)
Prove the conditionalized version of the general product rule:

$\displaystyle {\bf P}(A,B\vert C) = {\bf P}(A\vert B,C) \cdot {\bf P}(B\vert C) $

b)
Prove the conditionalized version of Bayes' rule:

$\displaystyle {\bf P}(A\vert B,C) = \frac{{\bf P}(B\vert A,C) \cdot {\bf P}(A\vert C)}{{\bf P}(B\vert C)} $

c)
Officials at the suicide prevention center know that 2% of all people who phone their hotline actually attempt suicide. A psychologist has devised a quick and simple verbal test to help identify those callers who will actually attempt suicide. She found that
  1. 80% of the people who will attempt suicide have a positive score on this test.
  2. but only 5% of those who will not attempt suicide have a positive score on this test.
If you get a positive identification from a caller on this test, what is the probability that he would actually attempt suicide?

d)
Imagine that you are a contestant in a game show and there are three doors in front of you. There is nothing worth having behind two of them, but there is $50,000 behind the third. If you pick the correct door, the money is yours. You choose door A. But before the host, Monte Hall, shows you what is behind that door, he opens one of the other two doors, picking one he knows has nothing behind it. Suppose he opens door B. This takes B out of the running, so the only question now is about door A vs. door C. Monte now allows you to reconsider your earlier choice: you can either stick with door A or switch to door C. Should you switch? Use Bayes' rule to answer this question.

e)
[1* P]

You watched the show in d) many times and noticed that Monte has a pretty bad poker face. He smiles in 30% of the cases where the money is behind the door that the contestant has initially chosen, but only in 10% of the cases where the money is behind the other door. How would this information influence your choice in d) if Monte would be smiling? Write down all your assumptions. (Hints: You may not be able to calculate the exact probabilities, but only their ratio.)


next up previous
Next: Conditional Independence [2 P] Up: MLA_Exercises_2007 Previous: Exercises
Haeusler Stefan 2007-12-03