# Estimation of the interval of a proportion (mark, recapture)

I'm a college studen doing statistics homework and Im looking for help. I have tried to solve this exercise, thinking about it in a number of ways, to no avail. This is the problem:

To estimate the number of frogs there are in a pond, we get a few out, 55 to be more precise, and we mark them and put them back in the pond. After a few days, we get a few out and we notice how many are marked. This last time we got 100 out of the pond and 7 were marked. Calculate an interval with 95% confidence for the proportion of marked frogs.

According to the aid of the professor, the answer is L1=0.07 and L2=0.19, which I'm not sure if it's even correct, because he only gave us the answer not the procedure.

My question is, what is the correct answer? I was adviced previously to use some methods to find the answer, but given my unfamiliarity with said methods, I'm afraind I may reach the wrong answer, and it would be great to know the correct answer to be able to study properly. Thanks in advance! (I added info according to guidelines of this exchange, if something is missing please don't close the question and allow me an opportunity to add possible missing information).

Below is the way I tried to work it out.