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).(adsbygoogle = window.adsbygoogle || []).push({});

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) = C_{N}(n1,N−n1) p^(n1) q^(N−n1), (2.16) where C_{N}(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 C_{N}(n1,N−n1) is C^{n1}_{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.

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# Two state system probabilities

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