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@teome
^^^^Yes but you're only considering a very specific example of equally probable events and the order matters. Others and myself were pointing out situations where the order is not important and often what is of interest is the frequency of occurance, not exactly when it happened. Probability Demsity Functions are based on the frequency of event divided by the total number of events and in physics we are often just interested in how many times. You're talking about the ways to arrange a system which is a part of it. I'm not arguing with your point but talking about more widely used probability.
If the runs were repeated infinitely many times all outcomes would occur. It's infinite! that's the basis of much of the theory, that for infinite runs all will occur
A typical example in physics is energy levels: two electrons e1 and e2 and two energy levels 0 and 1
There are four possibilities: e1_0 e2_0, e1_0 e2_1, e1_1 e2_0, e1_1 e2_1 but two give the same total energy. The expectation energy is
= 1/4 * 0 + 1/4 * 1 + 1/4 * 1 + 1/4 * 2 = 1
hence, the expected total energy is 1 and is the most probable outcome has probability 1/2 compared to 1/4 for 0 or 2 (arbitrary unitis). Much of classical statistical mechanics is based on the ways to arrange a system and calculations of the coefficients for a very simple 2 level quantum system. I/4 and 1/2 are the squares of the wavefunction coefficients. From the probabilities the entropy is calculated by
S = - sum [p_i * ln(p_i)] for all i