I haven't time to read all the posts, but here goes:

The question I answered was "So if that infinite set contains everything then what's the probablity of a given thing being contained in it?"

1, since the infinite set contains everything, so any given thing has 100% chance of being in the set.

The other confusion was around infinte games. The example of 2 pieces on each side being next to each other works like this (assuming player is on 5 spot and opp on 6 spot of his own home):

Game 1: 6-6 GAME OVER
Game 2: 1-1 1-1 6-6 GAME OVER
Game 3: 1-1 1-1 1-1 1-1 6-6 GAME OVER

etc.. infinite number of games.

The example where players continually hit each other is an infinitely long game, not an infinite NUMBER of games.