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Math Help - Probability Question

  1. #1
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    Probability Question

    Hi, can anyone help me out or give me a hint?
    "Abe owns an apple orchard. He is having pest problems, and he has a scanner at his disposal which allows him to check if a tree is infested or not. He knows that 25% of all trees are infested. The technology is correct 80% of the time. If he checks 100 trees at random, how many can he expect the scanner will determine to be pest-free?" This question is really confusing because there are 2 probabilities. My guess is that out of the 75 trees that are not infested in the random sample the scanner will say that 60 trees are pest-free and out of the 25 that we would expect to be infested the scanner would say 5 are not infested. So 65 trees is that right?
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  2. #2
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    Re: Probability Question

    Pr[infested] = 0.25
    Pr[!infested] = 0.75

    The probabilities of the scanner behavior is

    Pr[infested|infested] = 0.8
    Pr[pest free|infested] = 0.2

    Pr[infested|!infested] = 0.2
    Pr[pest free|!infested] = 0.8

    So Pr[pest free] = Pr[pest free|infested]Pr[infested] + Pr[pest free|!infested]Pr[!infested]

    Pr[pest free] = 0.2*0.25 + 0.8*0.75 = 0.65

    and so with a 100 tree sample we expect that 0.65*100 = 65 trees will register as pest free by the scanner.

    So you are correct, though you sort of stumbled to the answer.
    Thanks from mathbeginner97
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  3. #3
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    Re: Probability Question

    Thanks!
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  4. #4
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    Re: Probability Question

    When you solve the problem formally, you multiply the probabilities of the scanner scanning pest-free by the percentage of the trees that are pest free and infested and then adding them because then you can use the 0.65 to multiply any number of trees, right? like if 100000, 200, 15, etc. trees were scanned. The way I did it, I could only solve the problem for 100 trees.
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  5. #5
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    Re: Probability Question

    Quote Originally Posted by mathbeginner97 View Post
    When you solve the problem formally, you multiply the probabilities of the scanner scanning pest-free by the percentage of the trees that are pest free and infested and then adding them because then you can use the 0.65 to multiply any number of trees, right? like if 100000, 200, 15, etc. trees were scanned. The way I did it, I could only solve the problem for 100 trees.
    yes.

    0.65 is the probability that a given tree will be declared pest free by the scanner. So if you sample N trees on average you will have 0.65*N trees declared pest free.
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