Random allocation of coins and the exponential distribution

Suppose we have a fixed amount of money that we want to randomly allocate among some number of people. For example, let's say we have 20 dimes and 10 people to give them to. What is the most probable way to allocate the dimes at random?

The number of ways that N=20 dimes can be distributed among M=10 people is

N!/(n1! n2! ... nM!)

where n1 is the number of dimes given person 1, n2 is the number of dimes given person 2, and so on. The factorial N! = N x (N-1) x (N-2) x ... x 2 x 1. The sum of the number of dimes that each person has is equal to the total number of dimes: n1+n2 + ... + nM = N.

Since there are M people, there are M ways of allocating one dime. There are MxM=M2 ways of allocating 2 dimes. And there are MN ways of allocating N dimes to M people. The probability of having a particular random allocation of dimes is

probability(n1 ... nM) = (1/MN) N!/(n1! n2! ... nM!)

The probability of observing an equitable distribution, that gives the same number of dimes to each person, is fairly likely. For example, suppose that each of the 10 people receives 2 dimes. There are 20!/2!10=2.37 x 1015 ways to distribute the dimes equitably! The probability of observing that equitable distribution at random is

probability(n1=2 ... n10=2) = (1/1020) 20!/2!10 = 2.37 x 10-5

In contrast, the probability of observing a highly inequitable distribution, that gives many people no dimes and one person all the dimes, is far less. There is only one way to give person 1 all 20 dimes and everyone else no dimes. The probability of observing that most inequitable distribution at random is

probability(n1=20) = (1/1020) 20!/20! = 1 x 10-20

Random allocations will rarely lead to single individuals getting all the dimes. From the perspective of an individual, the probability of receiving a few dimes is very high, and the probability of receiving many dimes is relatively low.

It turns out that when the number of dimes and the number of people are large, the probability of finding an individual with a certain number of dimes is well approximated by an exponential distribution. The probability of having a little is exponentially larger than the probability of having a lot.

There are many circumstances where a fixed resource is randomly distributed among a large number of things. For that reason, the exponential distribution arises in many circumstances. In economics, if we have a fixed amount of money distributed randomly among a fixed number of people, we can expect an exponential distribution of wealth. Many people will be poor and a few will be wealthy. In the case of atoms and molecules, the situation is similar. A fixed amount of energy randomly distributed among a large number of molecules will lead to an exponential distribution of energetic wealth. Many molecules will have relatively little energy and only a few will be highly energetic!