I understand the limitations of the simulation. However, it indicates clearly that the chance to get knocked out after winning 34 games is very very close to zero.
It's so close that I would question whether both the derived formula and the sim results can both be correct.

The problem with the sim is it assumed all players were evenly matched.

The equation calculates the most extreme outcome, regardless of probability. Thus it is clear that 44 wins will guarantee a player at least a 16-way tie for 16th (and first) place.

The simulation shows that a more likely number around 31 to 33 will be good enough, but no guarantee.


I considered whether or not the 44 number would require collusion among the top 15 players and if a non-colluding 16th player could break that number, but the simple fact that it could happen with any possibility randomly means that it does not require collusion, and hero must win enough against the field to keep up.

Now, what will always happen is that half way through, hero can calculate a more reasonable magic number and control her fate by skill on the table.

If the top 15 players have a reason to include the 17th best player and collude to give a few extra wins to that player, that strategy would not increase the magic number for hero, but would make it a more firm cutoff, and hero would need to hit that number.

Yes I understand that. But 100% in the sim using 30 million coin flips, and poker edges aren't big. Doing the sim with some random edges might be useful but won't make a huge difference in the threshold to qualify.