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Math Help - Poisson Distribution Question

  1. #1
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    Poisson Distribution Question

    So I got a summer homework package from school , and I need to teach myself Poisson Distribution before school starts. So here's the question I've been stuck on for a while. (I need help on b.)


    The number of faults in glass sheets occur at a rate of 2.1 per sq. metre. If a 1x1 sq. m glass sheet contains at least 3 faults it is returned to the manufacturer.

    a) Find the probability that a 1x1 sq. m sheet is returned to the manufacturer.

    My answer:

    P(X > 2) = 1 - P(0) - P(1) - P(2)
    P(X > 2) = 1 - e^{-2.1}(1+2.1+\frac{2.1^{2}}{2})
    P(X > 2) = 0.3504

    b) Six such glass sheets are inspected. What is the probability that at least half of them are returned to the manufacturer? (Here's the answer provided: 0.6817)

    It would be nice if you can show me the work and proper notation, because I barely have a clue.

    Thanks.
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  2. #2
    Senior Member Danneedshelp's Avatar
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    Let Y be a random varibale that represents the number defective glass sheets. Then, Y is Binomial(n=6, p=0.3504), since the probability of a success is equal to the probability of a sheet of glass being defective. Thus, the probability that at least half the glass sheets will be defective out of a batch of 6 glass sheets ought to be

    P(Y\geq\\2)=\sum_{y=3}^{6}{{6}\choose{y}}(0.3504)^  {y}(1-0.3504)^{6-y}.

    I think that's correct.
    Last edited by Danneedshelp; August 2nd 2010 at 08:17 PM.
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  3. #3
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    Thanks for your reply.

    I see what you are doing using binomial distribution. However, the answer that your equation got wasn't the answer that was provided. The answer that you got was about 0.353.

    I also used the Poisson Distribution way to solve the question based on your logic, and it got the same (or approx.) answer:

    \lambda = (0.3504)(6) = 2.124

    so,

    P(X > 2) = 1 - P(0) - P(1) - P(2)
    P(X > 2) = 1 - e^{-2.124}(1 + 2.124 + \frac{2.124^{2}}{2})
    P(X > 2) = 0.3568

    Hm.. are we doing something wrong or the answer provided is just wrong?
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  4. #4
    MHF Contributor
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    Danneedshelp is correct,

     Y \sim Binomial(n=6, p=0.3504)

    P(Y \geq 2) = 0.681689

    You can use Binomial Distribution: Probability Calculator to check your answers when calculating binomial probabilities.
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  5. #5
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    Quote Originally Posted by timmcan View Post
    So I got a summer homework package from school , and I need to teach myself Poisson Distribution before school starts. So here's the question I've been stuck on for a while. (I need help on b.)


    The number of faults in glass sheets occur at a rate of 2.1 per sq. metre. If a 1x1 sq. m glass sheet contains at least 3 faults it is returned to the manufacturer.

    a) Find the probability that a 1x1 sq. m sheet is returned to the manufacturer.

    My answer:

    P(X > 2) = 1 - P(0) - P(1) - P(2)
    P(X > 2) = 1 - e^{-2.1}(1+2.1+\frac{2.1^{2}}{2})
    P(X > 2) = 0.3504

    b) Six such glass sheets are inspected. What is the probability that at least half of them are returned to the manufacturer? (Here's the answer provided: 0.6817)

    It would be nice if you can show me the work and proper notation, because I barely have a clue.

    Thanks.
    For (b), it looks that the answer provided is the probability when only 4 glass sheets are inspected.
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  6. #6
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    Thanks for the link SpringFan, but Danneedshelp used P(X \geq 3), not P(X \geq 2). Still, it seems that what you did came up as the correct answer according to the answer sheet. I guess there's just something wrong with the wording of the question.

    Thanks for your help.
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