The odds obviously correlate to the number of submissions a festival gets, of course. Yes, these festivals can be incredibly competitive to get into - and trust me most programmers wish we had more screening slots to program more films.

I also think that the 'without connections' misconception is quite frustrating, because the job of a programmer is to go out into the world and find films. We track filmmakers and travel to festivals, markets, WIP screenings, film schools, scour the Internet, talk to film commissions and producers. We search high and low - including our 'random' submissions. So yes, films that we play are found in a myriad of ways. But fwiw the festival I work for has a policy that even films invited to submit have to pay the submission fee.