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Math Help - Probability when drawing from a box

  1. #1
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    Probability when drawing from a box

    Hi again!

    I feel like this question should be simple enough, but I seem to be getting the answer wrong all the time:

    The problem:

    A merchant receives two batches of fish cakes (I live in Norway, we always use fish cakes) from the same producer but from
    two di ferent factories. Thus, from the outside the units of fish cake boxes are indistinguishable
    from each other. The batches are called A and B, and they have the following characteristics:

    Batch A: 100 boxes, whereof 10 with mushy cakes, described as F1.
    Batch B: 100 boxes, whereof 5 with mushy cakes, described as F1; and also 10 with foul
    smell and taste, described as F2, whereof 1 with both F1 and F2.


    Consider the errors mushy cakes and foul smell and taste to be impossible to detect from the
    outside, but can only be detected by destroying the inspected box.


    a) Draw venn-diagrams for each of the batches A and B, by regarding the number of boxes.
    If the merchant would draw a random box from batch B, what is the probability that
    he would draw one with the error F1? If he would draw two boxes randomly from batch
    B, what is the probability that he would draw one with error F1 only and one with
    error F2 only.

    What I did so far to solve it:

    Drawing the Venn diagrams was straight forward enough, and also calculating the probability F1, which was simply 5/100. However, when I had to calculate the probability of F1 and F2 I got it wrong. This is how I thought:

    The possible ways of drawing 2 boxes from a set of 100 is (100! / 2! x 98!). The possible ways of drawing one box with the error F1 is
    (5! / 4!), and the possible ways of drawing one box with the error F2 is (10! / 9!), so I thought the answer should be
    (5! / 4!) x (10! / 9!) / (100! / 2! x 98!), which is 0,01. However, this is wrong, the answer in the answer key is 0,00727.

    Can anyone explain to me what I am doing wrong?

    Thank you!
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Probability when drawing from a box

    Of the 5 cakes with F1 one also has F2. Hence there are 4 cakes that have F1 only.

    Of the 10 cajkes with F2 one also has F1. Henece there are 9 cakes that have F2 only.

    Try the calculation again, this time based on 4 F1-only cakes and 9 F2-only cakes.
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  3. #3
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    Re: Probability when drawing from a box

    Thank you so much for the help! I should really have noticed that some of the F1 will also have F2.

    I was wondering if you could help me a bit further with the question?

    The batches were put in a storage, and after a while the merchant forgot which batch was A
    and which batch was B, even though they were placed in two separate units. The merchant
    draws randomly three boxes from one of the units, and keeps in mind which unit he draws
    from. He considers it to be completely random if the unit is batch A or batch B. After opening
    the three boxes he observes that two of them are without errors, and one of them has error
    F1 only

    a) What is the probability of observing two boxes without error and one with error F1
    only. What is the probability that the boxes were drawn from batch A?

    I am a bit confused here because I can't know what batch the man drew from. I can calculate the probability of drawing F1 from batch A and from batch B, but I don't know which one he is choosing, if you understand what I mean?
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