Probability Question

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bluerhino

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Feb 19, 2012
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Lie Detectors: A polygraph (lie detector test) has a false positive probability of 0.35, that is,the probability that an innocent suspect is determined to be guilty is 0.35, and a false negativeprobability of 0.2, that is, the probability that a guilty suspect is determined to be innocent is 0.2.
Let Fi be the event that a suspect fails her ith polygraph, and G the event that the suspect isin fact guilty. Assume that F1,F2,... are conditionally independent given G and conditionally independent given Gc. Also, assume that 60% of suspects are in fact guilty.
  1. (a) Find P(F), the probability of a random suspect failing a polygraph test.
  2. (b) What is the probability of an innocent suspect failing her first polygraph but not her second
    one?
  3. (c) Are F1 and F2 independent? Provide both a numerical and an intuitive justification.
The question that I'm stuck on is (c).
 
Last edited:
The way I've approached the problem is to prove that P(F1 ∩ F2 ) = P(F1)*P(F2)

  1. I know that P(F1 ∩ F2 ) = P(F1 ∩ F2 | Gc)P(Gc) + P(F1 ∩ F2 | G) *P(G)
    = [P(F1 | Gc)*P(F2 | Gc)]P(Gc) + [P(F1 | G)*P(F2 | G)]P(G)
    = (.35)(.35)(.40) + (.80)*(.80)*(.60)
    = 0.433
    And That P(F) = P(F | Gc)P(Gc) + P(F | G)P(G)
    = (.35)(.40) + (.80)(.60)
    = 0.62
    = P(F1) = P(F2)


    But P(F1) * P(F2) = (0.62)(0.62) = 0.3844, which isn't equal to P(F1 ∩ F2)!

    However, intuitively, I feel that whether someone fails one polygraph test should have no affect on whether he will pass/fail the next one he takes. My intuition tells me that F1 and F2 SHOULD be independent, but I don't know where I went wrong with my math....

    Please help!     :(
 
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