to take out all the balls from the table into holes.
playing by the rules. (hiting only white ball).
is there any simulation of this question.
for “simplicity” we can assume the the hit power is average, but the direction is random each time.
Let’s consider the case with just a cue ball and one other ball.
If they’re randomly placed on a 7’ x 3.5’ table their average distance apart is going to be about 3’ or 36’’. A pool ball is about 2.25’’ in diameter, so the chance of the cue ball hitting the other ball with a random shot is about (22.25)/(2pi*36) or about 2%. If you take into account a bounce or two that probably can be increased to about 5%. So say only about 1 in 20 shots will result in the cue ball striking the other ball.
The table has a perimeter of 21’ and the pockets are about 4’’ wide. But a ball will only go in a pocket if it’s closer to the center, so lets say the effective size of the pocket is about 2". That means that if you roll a ball in a random direction, its odds of going in a pocket are about (62) / (2112) or about 5%. So if you hit the ball so that it rolls off in a random direction there’s about a 1/20 chance it will fall in a pocket.
1/20 * 1/20 = 1/400 chance that a random shot will result in in the cue ball striking the other ball to drop it in a pocket. On average about 200 random shots to drop one ball.
A pool table has 16 balls. Dropping the first ball has about a 1/25 chance with each shot because you have multiple targets, but each time you drop a ball the odds get worse. Using rough averages across the entire game, you’re probable looking at about 1500 random shots on average to clear the table.
My math skills are not even close to being sufficient to critique The Hamster King’s computations. 
But my reasoning thinks that his ‘numbers’ might be off a little. For instance, if the initial hit on the cue ball is completely random, the ball could just bounce back and forth across the table and not even strike any of the other balls. It just seems that 1500 is too low of a number. :dubious:
Or is it presumed that the cue ball will strike one of the other balls, every time it is hit? 
His computations implicitly assume a random distribution of balls at the start, not set up for the break.
The break doesn’t change matters much. There’s probably about a 1/5 chance that a random cue shot will hit the initial rack of balls. Once the break occurs, you’re into random placement territory. Adding a few extra missed shots at the beginning doesn’t have much effect on the 1500 shot estimate.
I’ll readily admit that my calculations are crude. It was just a rough back-of-the-envelope estimate, and I’d be happy to see someone come up with a more accurate number.
There’s a couple Matlab billiards programs available here.
Modifying one to have it take random shots and count the number of shots until the table is clear might not be too hard.
If you take skill out of the equation, it doesn’t matter. In a simulation, if you set friction to zero, all balls can find a pocket eventually in one shot that causes every ball to move. If that one shot doesnt touch a ball, then the answer is infinity.
If you do account for skill, look up stats on straight pool. My guess is that it averages about one shot per ball. The handful of shot that miss balance out with shots that sink more than one ball.
Why would you set friction to zero in the simulation?
Depending on the simulator it might maximize chance of a random pocketing to hit the cue ball very very hard and get high activity. From a practical standpoint I think this would increase the chance of a ball-off-the-table foul, or object ball bouncing off pocket cushion without falling. Do simulators consider these effects?
The hit power is very important. i mension “average power” for simplicity.
Hamster reasoning is good and made me thinking that we should define “average length” meaning average length that ball is traveling in every shot.
The average length is result of friction and energy preservation. but i still think that in average Hamster reasoning is good enough. and 1500 shots feels not too little and not too much so its sound good.