# How often do I need to draw until all balls in an Urn are of the same color?

Suppose there is an Urn with $n$ balls, $m$ being white and $(n-m)$ being black.

Now we draw $c, c < n$ balls - any white ball drawn will be colored black - then we put all balls back into the Urn and repeat.

How often do we need to draw until we expect all balls to be black?

//- More clearly: how often do I need to draw until 95% of the white balls are colored in black?

• Forget about the colors, there are $m$ special balls, and we repeat our draws until we have drawn each of them at least once. – vadim123 Nov 17 '14 at 18:59
• What is the exact meaning of "we expect all balls to be black"? Probability over $\frac12$? Probability one? – Joonas Ilmavirta Nov 17 '14 at 19:03
• @JoonasIlmavirta : I need a finite solution: e.g. when are 95% of the former white balls black. Sorry for being unclear – juq Nov 17 '14 at 19:18

## 1 Answer

The expected number of draws to choose a white ball given there are $m$ of them is $\frac{n}{m}$, and now we must choose one of the $m-1$ remaining white balls, etc. until there are none left so the expected number of draws is:

$$\sum_{i=1}^m\frac{n}{i}=nH_m$$