Probability of recapturing 5 tagged fish out of 100

Question

I'm a maths student and I've been stuck at this problem for some days now, and haven't yet worked out a way of solving it. The problem comes form the book; "Probability and statistics, a course for physicists and engineers" by Arak M.Mathai and Hans J.Haubold.

2.5.12- In a lake there are n fishes, 50 of them are caught and tagged. Later, in the same lake, 100 fishes are caught;

(i) What is the probability that exactly 5 out of theese 100 are tagged (Assume no reproduction between them)?

(ii) How can the total number n of fishes in the lake be obatined with this information (the information being the solution to problem (i))?

Answer 1

Capture-Recapture Method to estimate population size.

If there has been time between tagging and recapture for the fish to randomize themselves for random recapture then the proportion $50/n$ of tagged fish in the lake should be about the same as the proportion $x/100$ in the lake. You say $x=5,$ so solve for $n.$

Also, following the Answer of @stacker (+1), to use the hypergeometric PDF (or PMF): From dhyper in R, you can find the $n$ to maximize $P(X = 5)$ as follows:

n = 100:2000
p = dhyper(5,50,n-50,100)
n[p == max(p)]
[1] 999

To avoid the (presumably unlikely) outcome that there are $x=0$ tagged fish upon recapture, a slight modification is helpful. (Chapman method instead of Lincoln-Petersen.) For this and other insights, please do read the Wikipedia link suggested by @JMoravitz.

Answer 2

Assuming that you mean exactly 5 fish in (i) you can use Binomial distribution to calculate the probability depending on $n$. You should also include some sort of Indicator function here to exclude the possibility of $n < 100$. For (ii) the first comment will probably help you. Good luck!

Answer 3

If $X$ is the number of tagged fish in the 100 fishes $X\sim\text{Hypergeometric}(50, n-50, 100)$. The probability that 5 are tagged is

$$\Pr(X=5)=\frac{{50\choose 5}{n-50\choose 95}}{{n\choose 100}}$$

Then $n$ can be found by solving for $n$ in the above equation if the probability is known.