Given a set $U$ of people, there is a subset $S\subset U$ of infected people. You only know the size of $U$ but want to estimate $\dfrac{|S|}{|U|}$. However, you don't have enough time to test every single person $p\in U$ on the virus so you decide to only test a fraction $K\subset U$ of
all the people and thus try to approximate the ratio $\dfrac{|S|}{|U|}\approx \dfrac{|\{p\in K\mid p \text{ is infected}\}|}{|K|}$. Given that you obtain a ratio
$r\in [0.4\cdot |U|, 0.6\cdot |U|]$, you want calculate the probability that the actual ratio $\dfrac{|S|}{|U|}$ lies in the interval $[0.4\cdot |U|, 0.6 \cdot |U|].$
I read some procedures about this often involving the normal distribution, however, I'm not allowed to use the normal distribution for this task. How could I attempt to solve this problem?