I was going through old tests for a Stats I class and I came across this question.

A researcher looking for evidence of extrasensory perception (ESP) tests 500 subjects using a large stack of cards. Each card has, on one side, one of the following five symbols: (omitted). The researcher brings each subject into a room and, after shuffling the cards well, selects one from the stack (without letting the subject see it) and asks the subject to identify the symbol on the card. After repeating this experiment many times, the proportion of correctly identified cards for each subject is computed.

a) Suppose one person correctly identifies 18 out of 50 cards. Construct a 93% confidence interval for p= the true proportion of correct answers this person would have if he was shown an infinite number of cards.

Now C.I. seem fairly straightfoward, the problem here for me is that the standard error for a sample proportion is $\displaystyle \sqrt{\frac{p(1-p)}{n}}$, where p should be a p-hat. Now as n approaches infinity, the standard error naturally decreases and approaches 0. So what am I missing here?

EDIT: Teacher messed up wording