My intuition was that a Poisson distribution would model this scenario well we were trying to model counts and that the rate lambda would be related to the binding probability and so this was my initial answer. My interviewer asked me to develop this more and I developed a basic binomial model given the binding rate p and the number of potential binding sites n. This satisfied the reviewer and he noted that the Poisson distribution answer I initially gave was also correct but that he would have wanted me to apply the limit as n goes to infinity and np = lambda remains constant.