I go on a safari. Each day, I see various animals, and a keep track of all of my observations. Assume that each time I make an observation I see a single animal. At one point, I've made a total of 153 observations and I've seen 127 rabbits, 22 gazelle, 3 lions, and 1 tiger (4 different species). This makes me curious: what is the probability that the next animal I observe will be from a new species, i.e., not one of the 4 I've already seen?
Is there a way to get an estimate or bound on this probability? If not, is there a pragmatic way to get an estimate, perhaps making some additional assumptions that might potentially be reasonable in practice?
I'm not trying to estimate the total number of species, but just that the next observation is from a species I haven't seen yet. I feel like I might have read about this problem somewhere in a paper long ago, but I can't remember the details -- and now I've run across a practical situation where I need to solve it. It feels like it ought to be possible to make some deduction by applying exchangeability, but I can't manage to make the details work out.