Statutory remedies and recovery of profits/damages
In the case that Joe infringes copyright, the highest statutory penalty is $150,000, if the court finds willing infringement. The copyright owner may alternatively recover any profits Joe made. (17 USC §504)
Joe may be enjoined to stop offering these clips.
Criminal prosecution of copyright infringement
Because of the commercial nature of the copyright infringement it is possible to be criminally charged under 18 USC §2319, if the infringement is wilful.
A good summary of criminal prosecution of intellectual property infringement is the Department of Justice Manual on Prosecuting Intellectual Property Crimes.
Worst case scenario
Joe got an opinion from a lawyer telling him that his plan was certainly copyright infringement. People love these 25 second clips. Joe earns one billion dollars from selling these clips. The owners of the Big Bang Theory attempt to get in on the game, but everyone that would be interested already has an account with Joe. Joe is asked to stop and doesn't. Joe adds a copyright notice to each clip claiming personal copyright on the material. Joe been previously found guilty of violating 17 U.S.C. s. 506(a)(1)(c) in an unrelated matter.
- The copyright owner could recover all profits (1 billion dollars) that Joe made. (17 U.S.C. s. 504)
- The copyright owner could recover costs and attorney's fees. (17 U.S.C. s. 505)
- Joe is fined $2500 for the knowingly false copyright notice.
- Joe is sent to jail for 10 years and fined $250,000.
In my opinion, the copyright owner would sue for an injunction for Joe to stop doing this, along with maximum statutory penalty for wilful infringement of $150,000. Joe would agree to stop, and the copyright holder would withdraw their suit. (This is of course not assuming the worst-case facts that I hypothesized in the previous section.)
Regarding criminal prosecution, if Joe stopped, I believe criminal prosecution would be unlikely. In 1999, there were only twenty-six criminal copyright infringement prosecutions. In 2000, thirty-two. (Ref).
This reference is dated, but around the year 2000, there was approximately a 10:1 ratio of civil proceedings to criminal proceedings relating to copyright infringement. More than half of the criminal defendants had infringed at least $70,000 worth of property.