Suppose that moves are generated from two players in accordance with three rules: each move is chosen uniformly at random among places on the board ($19 \times 19$, $9 \times 9$, or $k \times k$ with small $k$ say) which are not occupied, one player can pass at most twice in a row, and a sequence of two passes from both players is considered an illegal move (so that consideration of the game does not coninue any further). What proportion of these random games when played to at most $n$ moves end in an illegal move (that is one player suicides while not simultaneously capturing a group or plays in a ko after the last player played it)? Is there a certain value of $n$ where almost all games of size larger than $n$ end with illegal moves?
If this question is better suited elsewhere, feel free to suggest closing/migrating it.
Edit: The conditions about passing are in place mainly to avoid very long games where little stones actually appear on the board. The same questions could be asked without passing occurring at all, or with some other set of rules on passing if convenient. Also, superko rules are being ignored here, but the question with or without these implemented might be considered depending on which is more convenient.