Goldfish in a lake It is known that many people dump their pet goldfish into lakes, when they want to get rid of them with no remorse!
We wish to calculate the number of goldfish in a small lake, in which there are several other fish of various species.
For this purpose we pick 20 goldfish and put them a permanent mark and then release them into the lake. After one day (and assuming there are no changes in the population of goldfish or other fish – the system is “closed”), we pick 30 goldfish, of which 5 are found marked. What is the probability that the total population of goldfish in the lake is from 115 to 125?

I have found that this is done by using the “mark and recapture” method, by which we calculate the expected population to be

$\frac{20*30}{5} = 120$
But how do we calculate the probability for it to be in the requested range?
Of course by intuition, I guess it must be close to 100%!
 A: Your question

What is the probability that the total population of goldfish in the lake is from $115$ to $125$?

is only meaningful from a Bayesian perspective.
Your experiment tells you that the population is at least $45$ as you know there are $20$ marked fish and at least the $30-5=25$ unmarked fish you found.
To use a Bayesian calculation, you need a prior distribution for the population, and this will affect your calculated posterior probability based on the observation.  For example using R

*

*with an improper prior probability that is constant (i.e. just looking at sums of likelihoods) you could try

    sum(dhyper(5, 20, (115:125) - 20, 30) * 1) / 
    sum(dhyper(5, 20, (45:10^6) - 20, 30) * 1)
    # 0.08099914


*

*with an proper prior probability of the population being  $N$ is proportional to $\frac{1}{N^2}$, you could try

    sum(dhyper(5, 20, (115:125) - 20, 30) * 1/(115:125)^2) / 
    sum(dhyper(5, 20, (45:10^6) - 20, 30) * 1/(45:10^6)^2)
    # 0.1072485

which suggests that your guess that the probability the population is in that range "must be close to $100\%$" is far too high. Even if you were absolutely sure there were in fact say between $100$ and $150$ fish in the pond, so replacing 45 by 100 and replacing 10^6 by 150, the calculated probabilities for the range would still be relatively low.
