In reality, you could easily eliminate passive checks from the game entirely. What we used to do back in the day was have all the players each roll a d20 20 times and I would keep those lists and any time I wanted a "secret roll" i would just take and strike off the next number on the list. It is effectively the same thing minus the "average result" part. Otherwise, you could just have a procedure of everyone making a Perception check every "turn" of movement while exploring, whether they need to make one or not.
Now I am thinking about why we want secret rolls or don't want the players to know the die roll results in the first place. Is it for immersion?