Each person has a disease with probability $p$ independently. Out of $100$ random persons, $20$ tested positive. what is the value of $p$? Assume that the disease can be tested accurately with no false positives.
A first glance suggests that $P(B(100,p)=20)=1$. But turns out this never holds for any $p\in[0,1]$. Also, another perspective leads to believing that since we were able to find $20$ infected people from a "random" set of $100$ people, what we have at hand is a uniform distribution, so $p=20/100$. Could anybody clarify?