Which one is true?

Question

In order to assess the number $N$ of individuals of a species of animal living on an island, we propose to adopt the capture-recapture method. For this we capture $800$ individuals. These individuals, which correspond to the proportion $p$ of the total number of individuals living on the islands, are marked and then released. $1000$ animals are subsequently captured, of which there are $250$ animals marked during the first capture. Which of the following statements are true?

• We have $0.20$ as estimation of $p$
• $p = \frac{800}{N}$
• We have $3200$ as estimation of $N$

I am not sure how to deal with this. Can you help?

EDIT:

I know that two of them are right, but I can say which ones?

Answer 1

The assumptions are that in each case, the animals captured constitute a random sample of the animals on the island, that whether an animal is captured in the second sample is independent of whether that animal is captured in the first sample, and that no animals are born or die between the two samplings. If there were $N$ animals in all, the expected number of marked animals in the second sampling would be $1000 \times 800/N$. Thus if the actual number is $X$, you might take $800000/X$ as an estimator of $N$.

Answer 2

If $X$ represents the number of marked inividuals we observe out of our sample of $1000$ individuals, then for $x\in\{0,1,.\ldots,800\}$ we have $$P(X=x|N=n)=\frac{{800 \choose x}{n-800 \choose 1000-x}}{{n \choose 1000}}$$ Here, $N$ represents the number of individuals in the entire population. Define the function $f$ on the domain $\{1550,1551,\ldots\}$ by $$f(n)=P(X=250|N=n)$$ I would use the value of $n$ that maximizes $f$ as an estimator of $N$.