You are given a 100*50 snooker table (felt area) and two balls of diameter 1. One ball is placed in the center of the table and the other ball is randomly positioned. What is the probability that I will be able to shoot this second ball directly into the top left pocket without touching the central ball? (Assume pocket has radius 1)

Say the pocket is at (0,0) the ball at (25,50) the opposite corner (50,100).
If the center of the ball hits between the two 'edges' of the pocket, it goes in.  The edges of the pocket are at (0,1) and (1,0).

Consider the near miss points for the ball.  What I want to know is where the center of a ball can be, and still go in.  Drawing a line roughly from the pocket to the ball has a slope of +2, so the two "near miss points" are each one unit away from (25,50) but along a line with slope -(1/2) which is perpendicular to slope 2.
The near miss points are approximately:
(25 - 2*sqrt(5)/5  ,  50 + sqrt(5)/5 )   and
(25 + 2*sqrt(5)/5 ,  50 - sqrt(5)/5 )

I say roughly, because what we really want is to trace a line from the left pocket edge to the left near miss point;  and from the right pocket edge to the right near miss point.  So the nmp's might be slightly different.

Anyway the total area of points where the ball can't go in, is the area of a 2 unit circle centered at (25,50), plus the area between those two lines and also between the circle and the far corner.

Or look at it this way: a five sided figure ABCDE plus half of a circle of diameter 2
A: left near miss point
B: right near miss point
C: right near miss line intersects far wall
D:  (50,100)
E:  left near miss line intersects far wall

Posted by Larry on 2006-02-12 09:43:58