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1999 Algebra #10: Tunnels Through a Pyramid

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June 21, 2012
geometry3D geometrypyramidanalytic geometryprobability

Problem Statement

Pyramid EARLYEARLY is placed in (x,y,z)(x,y,z) coordinates so that E=(10,10,0),A=(10,−10,0)E=(10,10,0),A=(10,-10,0), R=(−10,−10,0)R=(-10,-10,0), L=(−10,10,0)L=(-10,10,0), and Y=(0,0,10)Y=(0,0,10). Tunnels are drilled through the pyramid in such a way that one can move from (x,y,z)(x,y,z) to any of the 99 points (x,y,z−1)(x,y,z-1), (x±1,y,z−1)(x\pm 1,y,z-1), (x,y±1,z−1)(x,y\pm 1, z-1), (x±1,y±1,z−1)(x\pm 1, y\pm 1, z-1). Sean starts at YY and moves randomly down to the base of the pyramid, choosing each of the possible paths with probability 19\dfrac{1}{9}. What is the probability that he ends up at the point (8,9,0)(8,9,0)?
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