While I don't have any concrete figures, I would say that 99.9% of the time, of books that actually get made, a film agent has shopped it around. Let's rewind a bit to discuss why.
I've said here before, but I'll say it again because I think I have a lot of new readers: getting your book published is a very, very difficult task. Getting it made into a movie makes getting your book published almost easy. In order for it to hit your local cineplex, an almost serendipitous stream of events have to occur. Including (but not limited to): 1) a film agent has to agree to take it on. So after finding a book agent, you now have to be vetted even further...these agents take on even fewer project than lit agents, AND there are fewer of them out there, so...the odds are small. 2) A producer (or director or some sort of behind-the-scenes figure) has to want to option it. 3) A studio has to agree to give this producer money. 4) A script has to be developed that all parties agree on. 5) The studio/producers has to decide that despite steps 1-4, it is still worth their time to pay everyone involved their big payouts by greenlighting the project. 6) You have to overcome a wide variety of snafus throughout the process (including but not limited to: weak scripts, temperamental directors, temperamental actors, temperamental producers, studio bankruptcy, etc, etc, etc.)
Phew! And those are seriously just SOME of the steps that come to mind. There are about a dozen others.
Film agents, like book agents, act as a filter between authors and producers/studios. The best agents (and I count mine among them - I'm very fortunate to have her), have relationships with producers, studios, directors, etc, and know what they're looking for, in the same way that lit agents have relationships with editors. Sure, of course, someone could read your book and contact you and want to option the rights. Definitely. But the odds that they'd have all of the other linchpins in place to actually get the movie made? Probably not high. I'm not suggesting that Steven Spielberg doesn't read books and contact authors - he might (though again, I'm guessing it's not his standard way of finding material - he has a team, I'm sure, who is always actively looking), but this route is sort of climbing up and over a mountain when there is a tunnel that offers direct access. But yeah, that tunnel has a pricey toll and doesn't allow everyone to pass through.
Sorry for the bad analogy. Anyway, I wish that I had other news; I wish I could say, yes, I know a dozen writers who have been contacted by legitimate producers who have then not only paid them fairly (I'm not talking about these ridiculous options for basically no money) but have gone on to get the movie made...but I can't. In fact, I know very few writers who have sold movie rights to begin with. Some, sure, but most? No.
But readers, correct me if I'm wrong. What say you? Possible to get your movie adapted without a film agent?