# Oz’ Balls Problem: The Solution

A commentator named Oz proposed the following question: You have a box with n red balls and n blue balls. You take out each time a ball at random but, if the ball was red, you put it back in the box and take out a blue ball. If the ball was blue, you put it back in the box and take out a red ball.

You keep doing it until left only with balls of the same color. How many balls will be left (as a function of n)?

Peter Shor wrote in a comment “I’m fairly sure that there is not enough bias to get $cn$, but it intuitively seems far too much bias to still be $c \sqrt{n}$. I want to say $n^c$. At a wild guess, it’s either $c = \frac{2}{3}$or $c = \frac{3}{4}$, since those are the simplest exponents between $\frac{1}{2}$ and $1$.”  The comment followed by a heuristic argument of Kevin Kostelo and computer experiments by Lior Silberman that supported the answer $n^{3/4}$.