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Bernoulli, Poisson & Normal Probability

  1. Sep 12, 2007 #1
    [SOLVED] Bernoulli, Poisson & Normal Probability

    The problem statement, all variables and given/known data

    Every chocolate bar contains 100 squares, with 10% of the individual squares presenting a health hazard to people consuming them.

    (a) Using the Binomial, Poisson and Normal distributions, write down formulas for
    the probability that a single chocolate bar has at least 3 but no more than 7 deadly
    squares.

    The attempt at a solution

    For the binomial part, I've just done Pr(x=3) + Pr(x=4) +.... Pr(x=7). Just wondering if that's what you would do?

    I've done the same for the Poisson, using μ = 10.

    I'm stuck on Normal distribution though.
    I'm thinking of using
    [​IMG]
    but I don't know σ.
     
  2. jcsd
  3. Sep 12, 2007 #2

    EnumaElish

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    You can look at this as a Bernoulli for each individual square with p=0.1 and q=0.9; then variance = pq for each square. That would be the first approximation. Then you need to formulate a way around the problem that squares in a given bar are mutually dependent; e.g. if you know the first 90 are safe then you know that the remaining 10 are deadly. At least that's how I interpret your question.
     
  4. Sep 12, 2007 #3
    Hmm, what do you suggest? I realise it's not as simple as the bernoulli and poisson where you can just add up the individual Pr's....
     
  5. Sep 12, 2007 #4

    EnumaElish

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    When I re-read the question I realized that the question does not imply a dependence. If there can be as few as 3 deadly squares, then knowing that 90 are safe does not tell me the remaining 10 are deadly. This makes it much easier. Under binomial, mean = np = 10 and variance = npq = 9, which you can apply to a normal distribution.
     
  6. Sep 12, 2007 #5
    Yeah so std dev is 3, mean is 10
    I just slap it into the above formula? Initially I thought I should integrate from 3 to 7, but it seems as though I should be doing from 2.5 to 7.5? Does this sound correct?
     
  7. Sep 13, 2007 #6
    edit: im totally confused for normal dist now. Am I supposed to integrate at all or do I just use the formula I posted near the top? I mean to integrate that massive thing even though I know std dev and mean is out of my scope. Am I even on the right track?
     
    Last edited: Sep 13, 2007
  8. Sep 13, 2007 #7

    HallsofIvy

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    The normal approximation to the binomial distribution for given p (q= 1-p) and n has mean pn and standard deviation [itex]\sqrt{np(1-p)}[/itex]. Use those in your formula for the standard z-score, and find the probability that x is between 2.5 and 7.5. (That's the "integer correction")
     
  9. Sep 13, 2007 #8
    Can you guide me through it? I'm so damn lost.
     
  10. Sep 13, 2007 #9
    So basically...

    [​IMG]
     
  11. Sep 13, 2007 #10

    EnumaElish

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    Correct; except the left hand side is FX(7.5) - FX(2.5).
     
  12. Sep 13, 2007 #11
    Thanks a bunch!
     
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