I'm trying to understand why a poisson distribution may be preferred over a binomial one when modeling binary cases.
Is there a case where you can't use a binomial distribution to solve a problem that can be solved by a poisson distribution? For example: The average rate for fishing is 2 fish caught per day. What is the probability that 1 fish a day is caught? This can be modeled by a binomial distribution if you define a trial as each hour and the probability of success is approximately 2 fish/24 hour = 0.0833 per hour. Assuming there are 24 trials, the answer calculated from Binomial(24, 0.0833) is approximately the same as the poisson.
When is a binomial distribution unreasonable to calculate computationally? I understand that calculating a probability based on a poisson distribution is less computationally expensive than calculating it based on a binomial distribution. What are the practical applications of Poisson where computational time actually makes large impact?