The random variable we are going to study is
*X* the number of
white balls selected:

This only takes on positive values for:

Example:

Tagging animals, *N* unknown,
we conduct a tagging experiment
by catching and marking *m* animals and
recapture *i* of them that are tagged, out of the
recapture sample of size *n*.
We find it by maximising the probability
of
*P*(*X*=*i*)=*P*_{i}(*N*).

To see which *N* maximises this we notice
that :

This ratio is greater than 1 if and only if

This increases and then decreases and is max at floor(mn/i). This is what is called the maximum likelihood estimate of

Example: There are 50 tagged deer in the forest, mark them and release them, a subsequent catch n=40, of which 4 are found to have marks then .

Remark: If we supposed that probability of finding tagged animal is binomial, we get the same conclusion.