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Math Help - Probability Question

  1. #1
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    Probability Question

    A hall has n doors. Suppose that n people each choose any door at random to enter the hall.

    What is the probability that at least one door will not be chosen by any of the people.

    Here is my attempt:
    all possible combination: n^n

    P(at least one door not chosen) = 1- P(all doors chosen)
    = 1 - n!/n^n

    However, this is not what solution has, rather it contains

    (n-1)^n + (n-2)^n + ... + 1^n / n^n

    Wheres my flaw?
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  2. #2
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    I would agree to your solution ! Is there anyother information in the problem?
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  3. #3
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    Quote Originally Posted by Lukybear View Post
    A hall has n doors. Suppose that n people each choose any door at random to enter the hall. What is the probability that at least one door will not be chosen by any of the people.
    Here is my attempt: all possible combination: n^n
    P(at least one door not chosen) = 1- P(all doors chosen)
    = 1 - n!/n^n
    However, this is not what solution has, rather it contains (n-1)^n + (n-2)^n + ... + 1^n / n^n Wheres my flaw?
    There are many flaws with that approach.
    You need to count the number of surjections from a set of n to a set of n.
    That is simply n!

    So the answer is 1-\dfrac{n!}{n^n}
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