An urn contains $p$ black balls and $b$ white balls. A ball is taken from the urn and then the same ball is added back to the urn together with another ball of same color. This process is repeated indefinitely. Let $B_{n,k}$ be the event in which exactly $k$ white balls are taken in the first $n$ extractions. Show that:
$$ P(B_{n,k}) = \frac{{{b+k-1}\choose{b-1}} {{p+n-k-1}\choose{p-1}}}{{p+b+n-1}\choose{n}} $$
Hint: Show by induction on $n$, for all $p$ and $b$.
I'm completely stuck with this problem!
Can't even get the base case for induction where $n = 1$, that would be taking $k$ white balls with $1$ extraction. Why would that be $\frac{{{b+k-1}\choose{b-1}} {{p-k}\choose{p-1}}}{p+b}$?
Any help is highly appreciated. Thanks!