There are two kinds of coincidences: those that happen in real life, and those that happen in fiction. We're far more forgiving of the former. In fact, sometimes we marvel at the serendipity. In fiction, however, we tend to groan and roll our eyes when an author uses coincidence because it stretches our sense of credulity. And sometimes it's indicative of lazy writing. So, as a rule of thumb, use those serendipitous occurrences sparingly. However, like most rules, coincidence can be inserted into a story if it’s done in an artful fashion.
One of the reasons for this double standard is that in real life coincidences have much less significance and are more easily accepted. In fiction, if your plot hinges on a coincidence that is significant to the plot, it usually takes on a larger than life role. I remember back many years ago when I was a cop. I got a call of a suspicious person walking in a neighborhood and was sent to check the man out. I found him walking down the street, approached him, and asked to see his identification. The guy hadn't broken any laws and asked why I was stopping him. I gave him the usual explanation. As he was taking out his ID, I noticed he had a faculty identification card from Northern Illinois University, my old alma mater. I asked if he was from the university and he replied that he was a professor in the speech department. Years before, when I'd been an undergrad there, I'd dated a girl whose father was a professor in that same department. I asked if he knew this person and he gave me a startled look.
“Yes,” he said. “Why?”
I told him and he nodded, adding, “He died last winter, you know.”
I was shocked. I asked what had happened.
"It was during that big blizzard," the man said. "Heart attack."
Our conversation went on for a few more minutes as I inquired about the welfare of the deceased professor's family. The university was close to a hundred miles away from where we were, and I hadn't been there in years. In fact, I hadn't even thought about the place in several years. Now what were the chances that this professor, whom I'd never seen before, traveled all that distance to visit some relatives, who lived in the same area where I was a cop, and, coincidentally, that it would turn out that he and I shared a mutual acquaintance from that university. (Sure, that's a convoluted, run-on sentence, but those coincidences are complex.) Anyway, it was like six degrees of separation come to life. The irony struck my fancy, but did little else in the grand scheme of things. I was saddened, of course, but it certainly didn't result in a life-altering event for me.
In fiction, however, life runs at a different pace, and portrayed events have to be more significance. Minor coincidences, such as the one I described, wouldn't add much to the plot development. If I were to include such an event in a fictional story, I would have to make the occurrence mean something. Perhaps the deceased professor could have been murdered and the case was never solved. The visiting could be the killer, but he was stalking me because he knew that the motive was tied into my undergraduate days at the college. This is wretched plotting, I know, but you get the idea. The event would have to advance the plot in some way. Otherwise, the reader would be led down a path that went nowhere, and his interest would soon flag. We accept these whimsical chances of fate in real life, but they frustrate us when we're reading a story because they waste time and . That’s the difference between real life and its fictional counterpart.
This is not to say that a writer shouldn't use coincidence at all in his writing. Certainly, a clever twist of fate can greatly enhance a story, but it has to advance the plot. Thus, the insertion of some con incidence has to tie into the development of the plot. It can't be an unimportant or whimsical event. It's still going to stand out like a hungry ant on a powdered donut, so sometimes it doesn't hurt to draw attention to the coincidental event. Remember, it's going to stand out regardless, so why not tie it up with pretty bow? Sometimes subplots can be effectively used to make a coincidence more acceptable. In the movie Fuzz, based on one of Ed McBain's 87th Precinct novels, Detective Steve Carella is on a stake out as a homeless man when these two pernicious teenagers throw gasoline on him and light him on fire. Carella survives by jumping into a large puddle of standing rainwater, but is severely burned. These two punks are referred to periodically through the rest of the movie, but they don't reappear until the very end when they mistake the wounded villain for another bum and light him up. (I won't spoil the rest of the movie for you in case you want to see it. It's well worth it.) But you get the idea. The coincidence is worked into the story early on, so when it reappears at the end it doesn't seem like someone pulled a rabbit out of their hat.
Remember, the plot goes on, and, coincidentally, every word must advance it.