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Two state system probabilities

  1. Aug 13, 2015 #1
    I started to study statistical physics from a book, and it starts with basics about statistics and probabilities (which are things mostly new for me).
    In the book there is the following statement:

    "The simplest non-trivial system which we can investigate using probability theory is one for which there are only two possible outcomes. There would obviously be little point in investigating a one outcome system. Let us suppose that there are two possible outcomes to an observation made on some system S. Let us denote these outcomes 1 and 2, and let their probabilities of occurrence be
    P(1) = p,
    P(2) = q.
    It follows immediately from the normalization condition that
    p + q = 1,
    so q = 1 − p.
    The probability of obtaining n1 occurrences of the outcome 1 in N observations is given by
    PN(n1) = CN (n1,N−n1) p^(n1) q^(N−n1), (2.16) where CN (n1,N−n1) is the number of ways of arranging two distinct sets of n1 and N − n1 indistinguishable objects."

    I'm familiar with combinatorics, so I find it obvious that their CN (n1,N−n1) is Cn1 N.
    But I'm very curious how this can be proved. They give an example, but not a general demonstration. I've thought about it, but I couldn't do the demonstration. Can someone help me with it.
    I must mention: it is not homework, it is just for my personal knowledge.
     
  2. jcsd
  3. Aug 13, 2015 #2

    PeroK

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    You could prove it by induction. Have you come across mathematical induction?
     
  4. Aug 13, 2015 #3
    Yes, I know about mathematical induction, but I was never good at using it. After your reply, I tried to it, but with no success.
    First of all I tried to verify for 0, but it didn't verify, same for 1. I might miss something because I'm very tired.
     
  5. Aug 13, 2015 #4

    PeroK

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    If you're tired, why not take a fresh look at it tomorrow.
     
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