"W is sometimes called the "thermodynamic probability" since it is an integer greater than one, while mathematical probabilities are always numbers between zero and one" wiki

That is an odd usage. It’s like saying that the probability of rolling two dice whose sum is 11 is 2, rather than dividing through by the total number of equiprobable states to get 2/36, which is a proper probability.

Let me womansplain. The term "thermodynamic probability" is supposed to give an idea if a certain macrostate will occur in Nature. As an example imagine a box full of gas particles. Thermodynamic forces make the particles fill the box uniformly so that the same number of particles sits in the left and the right halves of the box. This is a macrostate. The n distinguishable particles can be rearranged in the box in [$]{n\choose 2}[$] ways/microstates and it's still the same macrostate. Another macrostate can be when one particle sits in the left half and the rest is squeezed in the right half - it seems uncanny, but there are still n microstates realising such a macrostate. Thus, thermodynamic probabilities of those macrostates are [$]{n\choose 2}[$] and n. The higher thermodynamic probability of the uniform macrostate means that it is more thermodynamically likely to occur.

This also means that a system converging to its equilibrium (which is defined as the macrostate with the highest thermodynamic probability - the uniform macrostate in this case) is maximising its entropy S, because S = k_B ln W (someone wrote this formula on Boltzmann's grave even if he never used it himself - yet another proof that physics is a lie). And this also means that the information we have about the system is decreasing towards the equilibrium, because when we observe the system we can see the macrostate only and not the particular microstate, which currently realises it. The more possible microstates, the less certain we can be which one makes up the macrostate. That's where information theory begins.

If you start to analyse the dynamics of the problem, you can ask which transition is more probable: from the first to the second microstate or vice versa? The difference of their thermodynamic probabilities gives you the answer, and eventually leads to defining reversible v irreversible processes... and the theory of ergodicity.

And so goes physics - sometimes it's not complete nonsense (or at least they managed to tweak it in such a way that it's hard to falsify - it's, as Cuchulainn's favourite goes, "not even wrong")...