PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles · Statistical foundations and the equilibrium ensembles
Thermal Physics, Ch Probability, random walks and the statistics of large numbers
Problem
A vessel holds N=1600 gas molecules, treated as independent points of negligible size, each of which is found in a marked region with a probability p equal to that region's fraction of the vessel volume. Two choices of region are compared: the region is half the volume (p=21), or a quarter of it (p=41).
((a)) For each p, find the mean count ⟨n⟩ and the standard deviation σn.
((b)) For each p, find the relative fluctuation σn/⟨n⟩, and say which region gives the sharper count in relative terms.
((c)) Using the Gaussian limit of the binomial, find the probability of the most likely count for each p, and explain why the narrower distribution has the taller peak.