Complexity of Games with Randomness

It is known that Hex is a PSPACE-hard game. (See this page, it is the "Hex ist PSPACE-vollstaendig" paper by Stefan Reisch.) Once we add a bit of randomness, however, the strategies suddenly become far easier! As I think I've mentioned before, if you change the game so that the current player is chosen by the flip of a coin, the winning strategy becomes simple.

I'm also interested in trying to analyze some games by removing their sources of randomness. Instead of relying on die rolls, for example, give each player the set of potential die rolls and let them choose one at a time, then removing that "roll" from their set of available moves. Once all options are chosen, either the player has no more moves or they get a new set of false rolls. I don't know of any analytical results like this, but I would love to see them!

On the other hand, I also have some interest in answering complexity questions of games that include randomness. For example: What is the probability that the current player has a winning strategy? Alternatively, rephrasing as a decision question, does the current player have a strategy that will allow them to win with probability > .5?

I don't know of any results like this. Do they exist?