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Homework Help: Binomial distribution

  1. Feb 28, 2013 #1
    1. The problem statement, all variables and given/known data

    Of all the weld failures in a certain assembly, 85%
    of them occur in the weld metal itself, and the remaining
    15% occur in the base metal. A sample of
    20 weld failures is examined.

    a. What is the probability that fewer than four of
    them are base metal failures?

    Is there a faster way to solve rather than doing p(x=0)+p(x=1)+p(x=2)+p(x=3)?



    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Feb 28, 2013 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    No, the way you said is about as short as possible. You can sometines speed things up a bit by doing it recursively: if
    [tex] P(k) = {n \choose k} p^k (1-p)^{n-k}[/tex]
    we have
    [tex] \frac{P(k+1)}{P(k)} = r(k) \equiv \frac{n-k}{k+1} \frac{p}{1-p},[/tex]
    so if we start from ##P(0) = (1-p)^n##, we can get ##P(1) = r(0) P(0),## ##P(2) = r(1) P(1),## etc.
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