It varies from one model to another, but what you are observing is quite normal.
When your oil pressure sensor goes bad, it will either peg to '3' all of the time (including when you just insert the key and turn it) or not function at all.
2009 ML350 (84K) - Family vehicle
2001 CLK430 Cabriolet (71K) - Wife's car
2005 BMW 645CI (124K) - My daily driver
2012 Mustang V6 (60K) - Daughter's car