Intel Mac :: 2011 Screen Unexpectedly Goes Black (to Sleep), Won't Wake

I'm using an iMac (late 2011) with 10.7.3. I'm doing things on the computer or some process is going on when the screen instantly goes black (no dimming first). If a video was playing, I can still hear it playing, but pushing keys on the keyboard or mouse or pad does nothing to wake the screen. Sometimes pushing the power button a number of times will get the screen to come on, sometimes it won't. When the screen goes black, if an external drive was running and then stops, I can sometimes get the screen to come to live. Sometimes this black screen happens when I'm burning a DVD; when the DVD is finished, I can bring the screen to life. I thought one of my two external drives was bad, so I disconnected it. That seemed to help (though verifying the disk revealed no problems). But a few minutes ago, the screen instantly went black again; no dimming first. There have been times I thought this was associated with a new bit of software, but then it returns even after I delete the suspect software. Sometimes I have to use the power button to shut down the iMac then restart it. Once, I had to unplug and replug the iMac before I could get the computer to respond and the screen to return to life. I don't know if this is a hardware problem (internal or external), a software problem, or some combination...or something I haven't thought of, at all.

Info:
iMac (27-inch Mid 2011), Mac OS X (10.7.3)