A probability always, always, always must be between 0 and 1. That scale means, "There is ZERO CHANCE AT ALL" for something to occur vs "There is ONE HUNDRED PERCENT CHANCE THAT THIS WILL OCCUR".
In basic principles, 0 means "It can't happen". So the idea of there being a negative probability does not actually have a rational meaning. 0 is already "No, it cannot occur". You cannot go lower than "It cannot possibly occur at all" in terms of likelihood.
Now, this is not to be confused with the very valid means of asking, "What is the probability that a negative outcome will occur?" or better yet, "What is the probability that I don't get x/y/z?" But those questions still lie between the 0 and 1 :)