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}.
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