Think of it this way: we can experimentally figure out what the probabilities are; they are an observed thing. The only way to make sense of these probabilities

Think of it this way: we can experimentally figure out what the probabilities are; they are an observed thing. The only way to make sense of these probabilities is if all protons are indistinguishable. In my statistical mechanics course, we went through an illuminating exercise where we started by trying to take account of every atom in a gas cloud. We started taking limits and making assumptions. One of them was that all atoms are indistinguishable from each other. This decreases the possible states of the system by N! (factorial, not surprise). After making that assumption, out pops the ideal gas law.