Question

(Graded for completion only) In countng problems, I like to divide cases up into three representative examples in which balls are placed in urns. Here are some ways of classifying these types of counting problems:

(a) Maxwell-Boltzmann statistics: Distinguishable balls are numbered 1,…k and placed into n urns. The sample space is the set of sequences (a₁,⋯,aₖ) with 1≤aᵢ ≤n. Thus S is the set of maps f:{1,⋯,k}→{1,⋯,n}. With n=365, this is used to model to birthdays occurring in class of size k.

(b) Bose-Einstein statistics: The balls are indistinguishable. Thus the sample space is just the number of balls in each urn: the set of (k₁,⋯,kₙ) such that ∑ᵢ kᵢ=k (we saw how many such solutions there were in Lecture 3 ).

(c) Fermi-Dirac statistics: The balls are indistinguishable AND no more than one ball can occupy a given urn. Thus the sample space S is just the collection of subsets A⊂{1,⋯,n} such that ∣A∣=k; it satisfies ∣S∣=( ⁿₖ).

Some of you may have seen these names in physics, as the scenarios they describe indeed correspond to the possible arrangements or particles (bosons, fermions, etc...). For the scenarios below, state which type of statistics above are most applicable and explain why in 1-2 sentences.

(a) The number of ways to fly k distinct flags on n different poles (can have multiple flags or no flags per pole)
(b) The number of ways to fly k indistinguishable (same color) flags on n poles (again can have multiple flags or no flags per pole)
(c) The number of ways 2 Bosons can occupy 2 states (atomic urns), where the Bosons are indistinguishable and all configurations are equally likely.
(d) The number of ways 2 Fermions can occupy 2 states, where Fermions obey the Pauli exclusion principle (meaning no two Fermions can be simulatenously in the same state.)
(e) The age of each student in our Math 425 class.
(f) The number of distinct human genomes, where a genome consists of 3 billion ordered base pairs (a base pair consisting of two letters from the set {A,C,T,G}).

Answer 1

For the scenarios provided: (a) Maxwell-Boltzmann statistics, (b) Bose-Einstein statistics, (c) Bose-Einstein statistics, (d) Fermi-Dirac statistics.

Step-by-step explanation:

(a) The number of ways to fly k distinct flags on n different poles can be modeled using Maxwell-Boltzmann statistics, where distinguishable balls (flags) are numbered and placed into urns (poles), allowing for multiple flags on each pole. The sample space is the set of sequences of flags on poles.

(b) The number of ways to fly k indistinguishable flags on n poles can be modeled using Bose-Einstein statistics, where the flags are indistinguishable and the sample space only represents the number of flags on each pole.

(c) The number of ways 2 Bosons can occupy 2 states, where the Bosons are indistinguishable and all configurations are equally likely, can be modeled using Bose-Einstein statistics with a two-state system.

(d) The number of ways 2 Fermions can occupy 2 states, where Fermions obey the Pauli exclusion principle, can be modeled using Fermi-Dirac statistics. The sample space consists of subsets of states that satisfy the exclusion principle.

(e) The age of each student in the Math 425 class is not a counting problem and does not fit into the statistics mentioned.

(f) The number of distinct human genomes can be modeled using combinatorics, but it does not fall into the statistics mentioned above.