Conditional Probability: Why the Formula Matches Intuition

The one-particle average of (3/2)k_B T holds for any N. What N changes is fluctuation, and how much noise you accept determines how many particles you need.

Latent heat keeps an ice bath at 0 °C, but air gaps and a thin water layer can waste that advantage. Here is how to set up the bath, and what nobody has measured yet.

Natural density says numbers below any fixed bound are negligible. Countable additivity says you still cannot draw one uniformly. Here is where the two separate.