An apparatus exists whereby a collection of balls is displaced to the top of a stack by suction. A top level (Level 1) each ball is shifted 1 unit to the left or 1 unit to the right at random with equal probability. The ball then drops down to level 2. At Level 2, each ball is again shifted 1 unit to the left or 1 unit to the right at random. The process continues for 15 levels and the balls are collected at the bottom for a short time until being collected by suction to the top.
What would be the exact probability distribution of the position of the balls at the bottom with respect to the entry position, which is arbitrarily denoted by 0?