I was doing a problem from a textbook. It said that there were 209 waste treatment facilities in the US and 8 of them treat hazardous waste on site it then said that if 10 were randomly sampled then:
What is The expected value of the number of facilities with on-site hazardous waste treatment and what is the probability of having exactly 4 of these facilities in the sample.
I looked at this and thought it was a binomial experiment but according to the solution manual, it ended up being hypergeometric. I am unclear as to why this is the case. Is it because they arent truly independent because we are picking from a maximum number of facilities? To be clear, i mean to ask if it would indeed be binomial if instead the question asked that P(on site)=8/209 and there are an infinite number of facilities. That is the only solution I can think of
Thanks
Your intuition is correct. The hypergeometric distribution arises when you're sampling from a finite population, thus making the trials dependent on each other.
However, if your number of trials is small relative to the population size, then the binomial distribution approximates the hypergeometric distribution because not replacing each item has a negligible effect on the conditional probability of success in each trial.