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Math Help - Bernoulli and Conditional Convergence

  1. #1
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    Bernoulli and Conditional Convergence

    The following problem is under the Bernoulli chapter in my book and it's about conditional convergence.
    A hospital obtains its flu vaccines:
    40% from A
    50% from B
    10% from C
    Amount of ineffective vaccines:
    3% in A
    2% in B
    5% in C

    The hospital tests 5 vials from each shipment. If at least one of the five is ineffective, find the conditional probability of that shipment's having come from C.

    I know how to solve the problem if the information in bold was not given. But I don't know how the information in bold changes the problem. Can any one explain ? How do I factor in them testing 5 vials.

    thank you
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    Quote Originally Posted by Hitman6267 View Post
    The following problem is under the Bernoulli chapter in my book and it's about conditional convergence.
    A hospital obtains its flu vaccines:
    40% from A
    50% from B
    10% from C
    Amount of ineffective vaccines:
    3% in A
    2% in B
    5% in C
    The hospital tests 5 vials from each shipment. If at least one of the five is ineffective, find the conditional probability of that shipment's having come from C.
    Suppose that F stands for a batch of five fails.
    Then P(FC) is the probability a batch fails from C.
    If a batch fails it must come from A, B, or C.
    So P(F)=P(FA)+P(FB)+P(FC)=P(F|A)P(A)+ P(F|B)P(B) + P(F|C)P(C) .
    From the given P(FC)=P(F|C)P(C)=\left[1-(0.95)^5\right](0.1).
    You want to find P(C|F).

    PS. I have assumed that the hospital tests five vials all from the same supplier.
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  3. #3
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    Quote Originally Posted by Plato View Post
    P(FC)=P(F|C)P(C)=\left[1-(0.95)^5\right](0.1).
    My problem lies here. Why is P(F|C) = 1-(0.95)^5. For me if we ignore the bolded information P(F|C) is 0.03. How does the bolded information lead to that answer ?
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    Quote Originally Posted by Hitman6267 View Post
    My problem lies here. Why is P(F|C) = 1-(0.95)^5. For me if we ignore the bolded information P(F|C) is 0.03. How does the bolded information lead to that answer ?
    Where does "P(F|C) is 0.03" come from? It is no where in the given. Is it?
    We know that 5% of the vials from C are defected.
    We know that 10% of all vials come from C
    The test fails if at least on vial is defected: 1-(.95)^5.
    In other words, not all five pass.

    But again what about P(F|C) is 0.03?
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  5. #5
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    I think I got it
    F|C : event that one of the five vials taken from C is defective.
    So P(F|C) = 1 - P(none of the vials are defective)

    I mean't 5% instead of 3% in my earlier post.

    Thank you
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  6. #6
    MHF Contributor matheagle's Avatar
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    and I so wanted to learn what conditional convergence meant in terms of rvs.
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