What role does chance play in the extinction of species?

I understand that evolution plays a role, either because a species evolves into something better adapted or fails to compete against another such species. I also understand that catastrophic events play a role, whether asteroid collision or human encroachment.

I have read that extinction rates on islands are greater than elsewhere. The reason given is the greater chance of a sequence of unfortunate events wreaking havoc on a necessarily smaller population. How big a factor are such events in general?

In most cases, plants and animals live in a delicate balance with their surroundings. They reproduce up to a certain carrying capacity and then go through cycles of population increase and decrease. Is it reasonable to say that by pure chance there will eventually be an unfortunate combination of circumstances that will push a species over the edge into extinction?

Please do not interpret this question in political terms. I am by no means making a case against the preservation of endangered species. I am just wondering if the precariousness of life must eventually catch up to all species.