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Math Help - Negative Binomial

  1. #1
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    Negative Binomial

    Ok, so when I try solved the attached problem I don't know how to come up with the right answer which is : 0.55858


    My way of solving the problem is the following:

    x=50 to infinity = 499(0.003)^2 (0.997)^498 + 500(0.003)^2(0.997)^499

    Do I need to something like-> (0.003)^2(0.997)^498 / (1 - 0.997) ?
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  2. #2
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    Hello, MathRules!


    The probability that a machine produces a defective item is 0.003.
    Each item is check as it is produced.
    Assume that these are independent trials.
    Compute the probability that at least 500 items must be checked
    to find two items that are defective.

    Answer: 0.55858

    If at least 500 items are checked to find two defectives,
    then the first 499 contained exactly one defective or no defectives.

    . . P(\text{1 d{e}f}) \:=\:{499\choose1}(0.997)^{498}(0.003)^1 \:\approx\: 0.33528

    . . P(\text{no d{e}f}) \:=\:(0.997)^{499} \:\approx\:0.22330


    Therefore: . P(\text{1 or 0 d{e}f}) \:=\:0.33528 + 0.22330 \:=\:0.55858

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