In which way is chance or coincidence used in Ethan Frome?
Chance does play a role in many of the characters' lives. First of all, chance brings the narrator to the small town, where he is able to learn about and tell the story of Ethan Frome. Then, chance brings Zeena into Ethan's life, through his parents' illnesses. Similarly, Zeena's illness brings Mattie into Ethan's life too. Those are all coincidences that play a large role in bringing the main characters together.
Another thing to look at are some key events of the story. It is rather coincidental that the ONE time the pickle dish is used by Mattie happens to be the time that the cat decides to jump up and knock it over. That is an unfortunate chance event that leads to the rather dramatic ousting of Mattie from the home, which in turn leads to their accident. It is by chance that they are sledding the moment that Mattie fatally decides to have them both end their lives through the sled. If they had been at the train station, or even riding in the sleigh, circumstances might have been very different. I don't think it was chance that they crashed; I think that Ethan steered them into the tree. The odds of survival chanced in their favor after that though. They are crippled; Mattie devastatingly so, and that circumstance leads to their imprisoned existence from there on out.
I hope those thoughts helped; good luck!