lean_workbook_plus_65724

Let $f(a, b)$ be the expected number of white balls left after all black balls get chosen, starting with $a$ white and $b$ black balls. Then $f(0, b) = 0$ , $f(a, 0) = a$ and $f(a, b) = \frac {a f(a - 1, b) + b f(a, b - 1)}{a + b}$ . This lets us compute as many values as we like, in particular $f(15, 10)$ . If we look at a few computed values, it also lets us conjecture that $f(a, b) = \frac {a}{b + 1}$ , which then follows immediately from the recursion by induction.