You exclaim "What the possible fuck!" when you aren't completely sure that this potential fucked-up situation is actually happening or has happened or ever will happen. In other words, you don't want to completely commit yourself to whether this is an actual fucked-up situation or not because you don't have all the facts. And if you misjudge it or call it prematurely, you would look like a dick. So meticulous and cautious soul that you are "What the possible fuck!" is the perfect reaction.
While watching fake-news BSNBC, Pete phones his friend Joe.
Pete (breathlessly excited): "MadCow says that a swarm of little green moon-men have landed their spacecraft on Times Square and and are handing out little green lollypops while shouting 'Fear not! We've come in peace!'"
A combination of possibly and probably. If possibly gives something a chance from 0% to 50% of happening, and probably gives something a 75% or greater chance of happening, then prossibly is the area between the two, the range of 50% to 75%.
Person 1: Hey, you want to go to the movies on Friday?
Person 2: Prossibly, depends on whether or not my boyfriend has to work.
Person 1: Dag, you can't let him control your life.
Person 2: But he loves me!
Person 1: Lame.