# A Variation of Banach Matchbox Problem

A Variation of Banach Matchbox Problem

We have two rolls of toilet paper. Assume that there are two kinds of people. Big-choosers always take a piece of paper from the roll that is currently larger, and little-choosers always take a piece of paper from the roll that is currently smaller. When the two rolls are the same size, or when only one roll is nonempty, everybody chooses the nonempty roll to which there is the smallest distance.

People come independently at random, with probability that they are big-choosers and probability that they are little-choosers. Initially, both rolls are of the same length. We are interested in the mean number of papers left on one roll when the other roll runs out. (Assume that everyone uses the same amount of paper and that the lengths of the rolls are expressed in terms of this unit.)

p

q=1-p

The Demonstration shows sample paths of the rolls and the mean number of papers left on one roll when the other roll runs out.