A "particle' is a term often used in physics for an object that, for the purpose of analysis, can be considered to exist at a single point in space i.e. that is effectively infinitely small. The mathematics we'll look at here is a lot simpler if these objects can be thought of as occupying no volume themselves.
There are a number of scenarios where we have a known number of "particles' distributed randomly through a finite quantity of some medium, and we want to model how many of those objects will appear in random samples from that medium. For example:
Vat of wine
How many bacteria in each bottle of wine drawn from the vat?
Diamonds of certain size
How many diamonds of a specific size in each piece of mined rock?
Newly sown field
How many seed in each m2 of the field
Let's use the bacteria in wine example. We'll say that the vat volume is V, the bottle volume is B, and that there are n bacteria in the vat. The probability that a bacterium will be in the first bottle drawn from the vat is the ratio B/V, so the number of bacteria s1 in that first bottle is:
s1 = Binomial(B/V, n)
Now, that leaves (n-s1) bacteria left in a volume (V-B), so the number of bacteria in the next bottle is given by:
s2 = Binomial(B/(V-B), n-s1)
And, using the same logic:
s3 = Binomial(B/(V-2*B), n-(s1+ s2))
Etc. In our software specific models, we have to add an extra bit of logic to ensure that the Binomial function does not return an error when the number of trials equals zero. Model Distribution of Particles shows an example.
Another example model: Particles in a Volume, calculates the joint probability distribution of the number of particles in both same-size and different-size samples. The total amount of particles in a volume is fixed, thus the probability distribution of the number of particles in a sample cannot be modeled as: s = Poisson(N/(V/n)), as the summation of particles in all samples will not be the fixed number.
It is interesting to note that in the case of equal samples, the probability distribution of the number of particles in each sample is exactly the same and does not depend on the order in which the samples are taken. That observation closely parallels a similar problem of hypergeometric sampling described here. In the case of different sample sizes, the mean number of particles in a sample is proportional to its size, which makes intuitive sense.
The links to the Distribution of Particles & Particles in a Volume software specific models are provided here: