Thank you j-astron for the warm welcome and your willing to help. I've read the rules of the forum. I'm PhD candidate and it's not my homework. Nevertheless, maybe I paid too less attention to my homework that is why I'm struggling this issue. I need this calculation to estimate how many companies I have to survey to fill my case studies grid. I have 8 cases
Ah ok. Well you've already managed to formulate a completely different problem that is mathematically-equivalent to yours, so I would say that you're already doing pretty well. By the way, in English, the past tense of "catch" is spelled "caught", not "couth".
I think that if we can solve this bird problem, it will tell you what you want to know, under the assumption that every company you survey is
equally likely to fall under any one of your eight cases.
I'm sorry if I'm not understanding your attempt at a solution, but it seems like what you're computing is the probability that each subsequent capture attempt will yield a new species of bird. However, that is a different problem from the original one, which is what is the probability that you'll get at least one of every species of bird after n capture attempts?
Based on the original problem, I think we can ignore cases of < 8 capture attempts. You cannot get one of every species of bird if you capture fewer birds than the total number of species.
I think we can solve this problem by doing the counting explicitly.
Probability = (number of desired outcomes)/(number of possible outcomes)
= (number of combinations that include one of every species)/(total number of combinations)
So we need to compute both the numerator and denominator (top and bottom) of this ratio.
For the bottom (tallying all possible outcomes): you are choosing k items from a set of n = 8 items, with repetition allowed. The order in which you receive the items is irrelevant. So this is
combination with repetition. Do you recall how to count combinations with repetition allowed?
For the top (tallying all desired outcomes): Right now I'm thinking it's
(number of ways of choosing one of every species) x (number of ways of choosing the remaining k - 8 birds)
= 1 x (8 choose k-8)
So for example, if k = 9, there is 1 way to select one bird of every species, so that we now have 8 birds. Then all we have to do is select who the
remaining (9th) bird is. There is only one remaining bird to select because in this case (k - 8) = (9 - 8) = 1. There are (8 choose 1) = 8 ways to choose who the remaining bird is. So the number of desired outcomes is 1 x 8 = 8.
So basically, compute the probability starting at k = 8 and incrementing by 1, until you reach a k value at which the probability exceeds 0.8.