December’s Geek Challenge is about trying to make pool shots on an infinitely large pool table.
To describe the challenge, let’s look at a 3x5 grid of pool balls with the cue ball positioned in the center. We want to know what the odds are that we can hit a ball chosen at random with the cue ball (without jumping or curving around other balls).
Looking at the possible cue ball paths, we see we can hit any ball except the 6 or the 9 because the 7 and 8 balls get in the way. Since there are 14 possible balls to pick from and we can hit 12 of them, there is a 6/7 chance we can hit a ball chosen at random from the 3x5 grid.
For the real challenge, we want to know our odds for an infinitely large grid of pool balls with one cue ball positioned like before.
Since this grid is substantially larger, the radius of the all the balls is now 0, so that a ball does not interfere with the cue ball path unless the cue ball hits that ball dead center. Because this is a much harder shot, we have also been granted the ability to hit the cue ball infinitely far and with perfect accuracy.
What is the probability that we could hit a ball chosen at random from the infinite grid without another ball being in the way?
Bonus points for the exact solution.
Please submit your responses to: email@example.com.
Check out previous Geek Challenges here!