Often a particular probability problem is not binomial at all. This happens when there is no repetition of equivalent experiments and/or there are more than two possible outcomes ("success" or "failure").
For example, imagine you have a small population of, say, elements, of which have a particular characteristic. From this population you take a sample of elements. would then be the number of elements in your sample that have the specific characteristic. The corresponding probabilities are:
.
The expected value is: .
If you take a small sample from a large population (for example out of , of which have a particular characteristic) you can still use the binomial probability model,
even though you are not actually dealing with equal chances. This is because the value
of the fractions and is almost the same.
In practice, when you take a sample from a large population, and you are interested
in the number of elements that carry a particular characteristic, then you can simply
use the binomial probability model.