I was thinking about this a little..well from experience of the past decade (mostly more tha 3 years ago): When high energy physicists impose their selection criteria they usually think by making such criterion very tight (strict cuts) they end up removing more background and by increasing the figure of merit they have a clean as well as better signal.
But I think one needs to make a trade-off here. One needs to loosen enough so there are plenty of back ground events. This has at-least two purposes.
One: it stabilizes the fit. By having less number of background events but not completely null one may have a problem making a stable fit.
Two: The back ground composition study with a slightly higher back ground events is a good thing to do. It shows what events are present SO one may do additional studies in hind sight that are not possible if there are no background events.
In any case I am saying if we understand the background events and can fit the signal and background well enough we have a better result than not having enough background events but ending up not studying them because they have been thrown away.
Correct me if I am wrong.