In a group of 1000 people, where each of them is either A, B, C, D or E (no one can be more than one thing). If I have exactly 200 people with each trait (20% per trait), when I take a random sample of 5 people, 1000 times with replacement (a bootstrap-like approach), I would imagine the percentage of these samples where at least one of the traits is not represented, is rather high, as it would be quite fortuitous to have randomly selected one of each. But what if I took a sample of 10 people? or 100?
Even when the distribution of traits is not homogeneous (e.g. A = 40%, B = 30%, C = 10%, D = 10%, E = 10%) or the traits distribution of the whole population is not available (eg. poll), how can we test what sample size is significantly representative?
My broader question is how big should my sample be to insure it has 95% chance of exhibiting the same proportions of traits in the group?
I have the impression that there is a "statistical/mathematical metric" between the number of traits, the size of our target population and how increasing sample sizes become more representative, but I'm missing the keywords to find it...