Using KRACH, I get a 20% probability that Cornell loses twice, and a 11% probability that Dartmouth loses twice. Those are pretty good unless at the same time, the result is that Cornell or Colgate wins the ECAC and claims one of the NCAA bids. Adjusting for that I'm seeing a net 25% chance that a Cornell or Dartmouth meltdown alone sends RPI to the tournament. But another strong combination is if Cornell beats Dartmouth, then loses to Yale (29%), there aren't many ways to beat that scenario.
If WM loses twice, that's a pretty certain ticket (38% chance).
It's looks tough to miss if UAA beats CC (47%).
Just from an hour of playing around, those look to me like the most important determinants. I confess that I did not look for any weird domino effects nor did I look at a Northeastern or UAA title run but they don't seem very likely. All that aside, the KRACH odds of at least ONE of those good things coming to pass and sending RPI to the NCAA is... 86%.
Check my math. Please carry forward all errors and give partial credit.