Poker hand probabilities in Doomtown:Reloaded
Creating a computer simulation is probably actually the easiest way to do this.

User 'Big Easier' created such a program and used it to create probabilities for the starter decks, which can be found here:

https://boardgamegeek.com/thread/1258645/draw-hand-simulator-preconstructed-decks

He mentioned putting the code on github, but I cannot find it.  Perhaps you could send him a message and get a copy of it.