Show and Change

Credit for this problem goes to a friend of Boon Leong Ng.

A ticket to a show costs 10 dollars. There are $(m+n)$ (where $m > n$) people waiting on a line. $m$ of them pays with 10 dollar bills and $n$ of them pays with 20 dollar bills. Assuming that the cashier starts out with no money, what is the probability that the cashier never runs out of change? Assuming the probability is uniform among all possible ordering of people.

Black and white painted sphere

The surface of a sphere is painted with black and white paints such that the area of black paint is less than 1/8 of the area of the sphere (the rest of the sphere is painted with white). Prove that one can inscribe a rectangular box in the sphere such that all eight corners land on white points.

More formally, if each point on the sphere is assigned a color black or white such that there is an injective but not surjective one-to-eight unordered mapping from the black points to the white points, prove that there are eight white points on the sphere that form a rectangular box.