1) The rain is not really random, since only when droplets get large enough do they drop to the earth. To get large they will accumulate whatever is closest around them, meaning a sheet of evenly-spaced vapor (due to the diffusive properties of gas) quickly becomes a sheet of larger droplets evenly spaced. And this even spacing falls to the ground - granted with some intervention by the wind etc but the point is the rain is not all that random.
2) The same thing happens on the surface of the table, right? Smaller droplets converge into larger ones as more rain falls and pushes them together. And there is a maximum size of dot before it turns into a lake and runs off the table. So eventually you get what tends to be a fairly even distribution of dots of a certain size.
Anyway I could be totally off-base with all of this, but let me just say this is a fascinating problem and what a curious mind to have come up with it!
2) The same thing happens on the surface of the table, right? Smaller droplets converge into larger ones as more rain falls and pushes them together. And there is a maximum size of dot before it turns into a lake and runs off the table. So eventually you get what tends to be a fairly even distribution of dots of a certain size.
Anyway I could be totally off-base with all of this, but let me just say this is a fascinating problem and what a curious mind to have come up with it!