Actually, I think exactly the opposite.  If there's any chance of failure, we want to catch it.

If there are many options, but we can't test them all, and we have no reason to choose one over others, then stochastic tests are the way to go, to get test coverage.  After all, maybe there's a weird youtube ID (or one that's accidentally blank, etc).

The keys are:
* Good reporting of the settings on a failure.  Need to report the video used.
* Knowledge that we're using stochastic tests
* A testing culture where people investigate tests on failure, not simply re-run them.