Was Herbert's death by fate or chance in "The Monkey's Paw"? What are some examples from the story to prove this?
I'm not so sure that the story was meant to be a horror story, primarily. Certainly a horrible thing happens in the story, the mangling to death of poor young Howard, but the story is just as philosophical as it is a tale of horror.
After all, much of the horror is in the mind of the reader, for no conclusion can be drawn as to why Howard died. And did Howard's disfigured, bloody corpse claw its way out of the grave, trudge all the way to his parents' home and knock on there door in the dark of night because of a wish? Or was it but the wind and your imagination?
Chance or fate? Who's to say? And how can one hope to prove one's theory one way or the other?
A wish was made. A death occurred. Money was obtained. There was a late-night knocking at the door. So?
Considering the timing and repercussions of each of the three wishes, I would say that Herbert's death was certainly fate. The story is meant to be a horror story, so - unless the author intended irony, which seems implausible - every event that occured is fated.
Of course, more questions arise from yours, such as, Was the knocking at the door chance or fate? etc. Keeping the genre in mind, a safe answer would be FATE, although you could find arguments for chance if you dig deeply enough. That, however, would be going out on a limb, and that would probably not be the author's intention anyway.