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Poisson approximation distribution

  1. Feb 19, 2012 #1
    1. The problem statement, all variables and given/known data
    In a manufacturing process for electrical components, the probability of a finished component being defective is 0.012, independently of all others. Finished components are packed in boxes of 100. A box is acceptable if it contains not more than 1 defective component. Justify the use of a Poisson approximation for the distribution of the number of defective components in a box. 8 boxes are randomly chosen. Given that all of them are acceptable, estimate the conditional probability that they contain exactly 6 defective components altogether.

    2. Relevant equations



    3. The attempt at a solution
    Let X be the number of defective components in a box containing 100 components, then X~B(100,0.012) => X~P(1.2)
    P(X<=1) = 0.663
    Let Y be the number of acceptable boxes in 8 randomly chosen ones, thus Y~B(8,0.663)
    P(Y=8) = 0.663^8 = 0.0372
    P(required) = P(8 acceptable boxes contain 6 defective components) / P(8 acceptable boxes)
    I could calculate the probability that 8 randomly chosen boxes contain 6 defective components (using a Poisson distribution with mean=9.6, I guess), but I'm clueless when it comes to '8 acceptable boxes'.
    Any feedback would be highly appreciated. Thanks!
     
  2. jcsd
  3. Feb 19, 2012 #2

    vela

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    You know that the eight boxes are all acceptable, so each has either one or no defective components. So six have to have a defective component. The remaining two have none. What's the probability of that occurring?
     
  4. Feb 20, 2012 #3
    So, P(X=1)=0.361 , P(X=0)=0.301
    P(8 acceptable boxes contain 6 defective components) = (8C6)((0.361)^6)((0.306)^2))=0.0056
    P(required)=0.0056/0.0372=0.151

    Am I getting it right?
     
  5. Feb 20, 2012 #4

    vela

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    Yup, that's it.
     
  6. Feb 20, 2012 #5
    That helps a lot. Thanks
    Oh, I have one more question here. If X and Y are 2 independent events, are X and Y', X' and Y' independent also?
    (X' and Y' are the complements of X and Y, respectively.)
     
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