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Math Help - grouped probabilities

  1. #1
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    grouped probabilities

    Probability question combinatorial mathematics?

    Eight different book, three in physics and five in electrical engineering, are placed at random on a library shelf. What is the probability that the three physics books are all together?

    I have the answer; its 3/28
    i have no idea how it is actually 3/28 anyone wanna clarify this?

    thank you
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  2. #2
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    Re: grouped probabilities

    Hey lhurlbert.

    If all the three books together then it means that you can look at each combination as "sliding" along for each possibility.

    You have 8 books in total which means if you start all books at the left-most side you get 6 possibilities where the books start at positions 1,2,3,4,5 and 6.

    Now we need to find the number of possibilities of re-arranging the books and for this we use a standard combinatoric identity.

    The number of ways arranging 8 books given 5 engineering books (or 3 physics books) is 8C5 = 8C3 or using R:

    > choose(8,5)
    [1] 56
    > choose(8,3)
    [1] 56

    So the total number of possibilities is 6/56 = 3/28 as expected.
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  3. #3
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    Re: grouped probabilities

    Hello, lhurlbert!

    Eight different book, 3 in Physics and 5 in Electrical Engineering, are placed at random on a shelf.
    What is the probability that the three Physics books are all together?
    Answer: 3/28

    There are: . 8! = 40,\!320 possible orders.


    Duct-tape the 3 Physics books together.
    They can be ordered in 3!=6 ways.

    We have 6 "books" to arrange: . \boxed{ABC}\;D\;E\;F\;G\;H
    They can be ordered in 6! = 720 ways.

    Hence, the 8 books can be ordered in 6\cdot 720 = 4,\!320 ways.


    Therefore, the probability is: . \frac{4,\!320}{40,\!320} \;=\;\frac{3}{28}
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