Equilateral Triangle Billiards 2

Probability Level 3

You are playing pool on a billiard which is equilateral triangle A B C ABC in the Cartesian plane, the coordinates which are A = ( 1 , 0 ) , B = ( 1 2 , 3 2 ) , C = ( 0 , 0 ) . A=(1,0),\quad B=\left(\frac12, \frac{\sqrt{3}}2\right),\quad C=(0,0). The ball is at point ( 7 10 , 3 10 ) \Big(\frac7{10}, \frac{\sqrt{3}}{10}\Big) in the triangle, and you hit it in any direction at random with uniform probability.

What is the expected distance the ball travels before hitting side A B AB , to 4 decimal places?


Details and Assumptions:

  • The ball bounces off the sides of the triangular billiard according to the law of reflection.
  • The ball is infinitesimally small.


The answer is 0.7412.

Christopher Criscitiello by Christopher Criscitiello , 25, USA

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1 solution

Hint: See link text . Tile the plane as in that problem.

Remark: I may have made a calculation mistake. If so, speak up! Thanks!

Below is a graph of the expected distance traveled on the z-axis, for each starting point in the triangle. As you would expect, the expected distance is smallest near AB, and approaches 1 near C.

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