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Homework Help: Help needed with statistics, probability

  1. Feb 16, 2010 #1

    rock.freak667

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    URGENT help needed with statistics, probability

    1. The problem statement, all variables and given/known data

    1)
    A plane functions iff at least 2 of its 3 engines function.
    P(each engine functions)=p, the engines operate independently of each other. Find the

    probability that the plane functions.

    3. The attempt at a solution

    A=engine 1, B=engine 2, C=engine 3.

    P(plane functions)= P(AuB)+P(AuC)+P(BuC)

    P(AnBnC)=p3

    P(AnB)=P(AnC)=P(BnC)=p2

    P(AuBuC)=P(A)+P(B)+P(C)-P(AnB)-P(AnC)-P(BnC)-P(AnBnC)=3p-3p2-p3

    http://img168.imageshack.us/img168/2991/venn.jpg [Broken]

    But the regions they want are P(AuB)+P(AuC)+P(BuC) which is just 3(p-p2-2p3)



    1. The problem statement, all variables and given/known data
    2)

    2% of products in a factory are defective. Products are sold in packages of 100. What

    proportion of cartons contain at least 'x' defective products? x=1,2,3,...,100. (use

    binomial distribution)


    2. Relevant equations

    Binomial distribution: P(X=x) = nCx px(1-p)n-x , where p is the probability of success.

    3. The attempt at a solution
    P(defective)=0.02
    P(not defective)=0.98

    I need to get X~Bin(100,0.98)

    P(X=x)= 100Cx0.98x0.02100-x

    So my answer is just P(X=x)/100 ? (replacing P(X=x) with the above)
    1. The problem statement, all variables and given/known data

    3)
    A shipment of 8 items contain 3 that are defective. A person makes a random selection of 2

    of these items, find the probability distribution for the number of defectives X. Find the

    cumulative functions of X as well.


    3. The attempt at a solution

    So P(X=x) is defined as follows:

    0 for x<1

    3/8 for 1<x<2

    5/56 for 2<x<3 ( 2 defective is 3/8 * 2/7)

    6/336 for x>3 (3 defective is 3/8 * 2/7 *1/6)

    And to get the cdf I just integrate the functions (in the regions) between 'x' and -infinity?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Feb 16, 2010 #2

    vela

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    Re: URGENT help needed with statistics, probability

    [itex]P(A \cup B)[/itex] is the probability that at least one of engines A and B is functioning, but you want the probability that both are functioning at the same time. Same problem with the other pairs.

    Also, you should have

    P(AuBuC)=P(A)+P(B)+P(C)-P(AnB)-P(AnC)-P(BnC) + P(AnBnC)

    not a negative sign on the last term.
     
  4. Feb 16, 2010 #3

    rock.freak667

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    Re: URGENT help needed with statistics, probability

    I typed the formula wrong :blushing:

    Then I would get P(AnB)+P(AnC)+P(BnC)=3p2, if that is the case, I don't understand why they told me to use that formula.
     
  5. Feb 16, 2010 #4

    vela

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    Re: URGENT help needed with statistics, probability

    Now you're triple-counting the intersection of A, B, and C, so you need to subtract the extra from your result. You can just add up the probabilities you have in your Venn diagram from the relevant regions to see what the answer should be.
     
  6. Feb 16, 2010 #5

    rock.freak667

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    Re: URGENT help needed with statistics, probability

    It would just be the intersections alone, so it'd be 3(p2-p3)+p3
     
  7. Feb 16, 2010 #6

    vela

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    Re: URGENT help needed with statistics, probability

    Yes, that's right. I see what they wanted you to do now. The plane functions if (A and B) or (A and C) or (B and C). Do you see how that's the union of 3 sets, so that that formula would apply?
     
  8. Feb 16, 2010 #7

    rock.freak667

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    Re: URGENT help needed with statistics, probability


    Wouldn't that give a different answer to the post above?

    P(AuBuC)=3p-3p2+p3, I'd have a 'p' term here.
     
  9. Feb 16, 2010 #8

    vela

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    Re: URGENT help needed with statistics, probability

    All good until the last line.

    If P(X=x)=0.10, that means 10% of the cartons have x defects. There's no need to divide by 100. The probability is already the proportion.

    Also, the problem isn't asking for P(X=x). That would be the probability that a carton has exactly x defective parts. The problem asks for the probability that a carton has at least x defective parts.
     
  10. Feb 16, 2010 #9

    vela

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    Re: URGENT help needed with statistics, probability

    No, because the three sets aren't A, B, and C.

    Both engines A and B working corresponds to [itex]A \cap B[/itex], and so on. (A and B) or (A and C) or (B and C) corresponds to the union of those intersections.
     
  11. Feb 16, 2010 #10

    rock.freak667

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    Re: URGENT help needed with statistics, probability

    So then if I find P(X=x-1), I'd get exactly 'x-1' defective, and then if I put 1-P(X=x-1), I should get 'x or more defective', which would be at least x, right?
     
  12. Feb 16, 2010 #11

    vela

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    Re: URGENT help needed with statistics, probability

    No, that would give you the probability of 0, 1, ..., x-2 defects or x, x+1, x+2, ... 100 defects. You only want the second part, not the 0 through x-2 part.
     
  13. Feb 16, 2010 #12

    rock.freak667

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    Re: URGENT help needed with statistics, probability

    Oh now I get it.

    This part has me confused :confused:
     
  14. Feb 16, 2010 #13

    vela

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    Re: URGENT help needed with statistics, probability

    1-P(X=x) is the probability of not having exactly x defects. That means you could have more or less than x defects, which isn't the same as having at least x defects.
     
  15. Feb 16, 2010 #14

    rock.freak667

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    Re: URGENT help needed with statistics, probability

    The problem wants me to get P(X≥x), I have P(X=x). I need to get 1-P(X<x). How would I get P(X<x)?
     
  16. Feb 16, 2010 #15

    vela

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    Re: URGENT help needed with statistics, probability

    You would sum the probabilities P(0)+P(1)+P(2)+...+P(x-1).
     
  17. Feb 16, 2010 #16

    rock.freak667

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    Re: URGENT help needed with statistics, probability

    I know it would be a binomial expansion, but of what? The normal bin distribution is (p+q)n where q=1-p
     
  18. Feb 16, 2010 #17

    vela

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    Re: URGENT help needed with statistics, probability

    It'll be part of a binomial expansion. As far as I know, there's no closed form expression for it.
     
  19. Feb 16, 2010 #18

    rock.freak667

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    Re: URGENT help needed with statistics, probability



    [tex]P(X \geq x) = 1 - \sum_{x=0} ^{x-1} ^{100}C_x 0.98^x 0.02^{100-x}[/tex]

    there is no way to simplify that term?
     
  20. Feb 16, 2010 #19

    vela

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    Re: URGENT help needed with statistics, probability

    Not that I know of.
     
  21. Feb 16, 2010 #20

    rock.freak667

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    Re: URGENT help needed with statistics, probability

    Then I have to leave my answer like that then.

    any help for the 3rd one?


    EDIT: I got out the 3rd one.
     
    Last edited: Feb 16, 2010
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