What is the exact probability distribution of the position of the balls at the bottom with respect

Question:

What is the exact probability of the position of the balls at the bottom with respect to the entry position (arbitrarily denoted by 0)?

SIMULATION

An apparatus displaces a collection of balls to the top of a stack by suction. At the top level (Level 1) each ball is shifted 1 unit to the left or 1 unit to the right at random with equal probability (see Figure 5.23).

Figure 5.23 Apparatus for random displacement of balls

Level : etc. etc. 15 -1 etc. ... ... etc. go00o 3. poooo 0000- N go000 0o000 0000- do000 o000000- C 00000 do000 do000 00

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; the balls remain at the bottom for a short time and are then forced by suction to the top. (Note: A similar apparatus, located in the Museum of Science, Boston, Massachusetts, is displayed in Figure 5.24.)

Figure 5.24: Probability apparatus at the Museum of Science, Boston

PROBABILITY

Distribution
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