MTtoCA: I'm pretty sure your assumption is correct. So cutting down your max size would slightly raise the chances of your getting closer to your first choice.

But this kind of situation only occurs when the slots are nearly filled during the lottery award processing. What it mostly depends on is where in the process your application is run through. If it is early in the process, your first choice will get the slots. Past the mid-point, all but the least desirable slots are filled, and none of your choices will win, even with the minimum group size.

When your application is run is completely dependent on the random order selection process.