A Probabilistic Approach to the Science of Dirty Rooms

Before beginning any serious discussion on the physics of dirty rooms, we are going to do a simple thought experiment. Consider the following scenario:

A game is played by throwing small ping-pong balls on a N x N large chess board. Assume that the balls will land entirely within a single tile. Don't necessarily assume that at most one ball will land on any single tile.

Consider the case where x balls are thrown (Where x<N^2). Notice that 1/N^2 is the probability that a single ball will land on some square. Since there are x balls and it is assumed that the placement of a previously thrown ball does not determine that placement of later balls, the probability of seeing any single configuration of balls is simply:

(1/N^2)^x.

For large values of N, given x>0, notice that this probability is approximately zero. Consider the probability that every single ball will land on one tile. Since there are N^2 different tiles, the chance of this occurring is

N^2(1/N^2)^x
=(1/N^2)^(x-1).

For large values of N, again this probability nears zero for x>1. The probability that this will not occur is simply 1-(probability that it will occur), or

1-(1/N^2)^(x-1)
=(N^(2x-2)-1)/(N^(2x-2)).

For large N, observe that the probability that every single ball will not land on the same square is approximately one. Using similar lines of reasoning, it can be demonstrated that the balls will tend to distribute themselves across the entire chess board as oppose to being stacked up on a single tile.