The following is the derivation of the probability of the most probable macrostate for the binomial distribution provided by Larry Caretto after I pointed out to him that this probability would become vanishingly small for large N. While he acknowledged that this had not previously occurred to him, he quickly responded with the following:
Derivation of P(most probable macrostate) = Wmp / Wtotal 6/14/13
In general, for N coin flips, the number of equally likely outcomes is Wtotal = 2N. From probability theory we know that the number of occurrences of any particular observed outcome is given by the following equation:
The most probable distribution, the one that maximized W(nH,nT) occurs when nH = nT = N/2.
We can now apply Stirling’s law (http://en.wikipedia.org/wiki/Stirling%27s_approximation) for the factorial of large numbers (valid since we will be considering a conceptual system of particles that we can make any size):
This gives
The basic idea is to compare Wtotal = 2N and Wmp for very large N. As you can see from the equation above, for this coin flip example
