Formula for urn model with replacement and "color switching"

Question

I tried to figure out following problem, but failed :/ My best guess is that it somehow is a mixture of a binomial and a hypergeometric distribution:

I have an urn with N white balls and 0 black balls. Every time I draw one white ball, I replace it with a black ball. A drawn black ball stays black (and is put back in the urn). I'm particularly interested in the mean of white balls in the urn after n draws.

Cheers, Andreas

Answer 1

This is Coupon Collector's problem, which could also be viewed as sum of $n$ iid Geometric rvs. Mean time until the box has only white balls is exactly $n H_n = O(n \log n)$.

Answer 2

• The probability that a particular ball is not drawn in one draw is $\dfrac{N-1}{N}.$

• The probability that a particular ball is not drawn in $n$ draws (i.e. that a particular ball is white after $n$ draws) is $\left(\dfrac{N-1}{N}\right)^n.$

• The expected number of white balls after $n$ draws is $$N\left(\dfrac{N-1}{N}\right)^n.$$