The beta-binomial distribution characterizes the number of successes in $n$ trials, but where the probability of success at each trial is unknown or random. However, suppose that you had finite populations and required that samples occurred without replacement (hypergeometric distribution).
Is there an analogue to the beta-binomial distribution which characterizes sampling without replacement from a population of $n$ items, where $b$ are labelled black and $w$ are labelled white ($n = b + w$), but where you do not precisely know what the values of $b$ and $w$ are?