Searching for Secret Doors

In my D&D group, I'm often disappointed by the lack of team work in searching.  Since the search roll is executed in secret, the fact that you did not find a secret door does not imply that there is none there.  When someone chooses to search I always roll a die in secret, even if I know there's no door.  That creates uncertainty. 

So.. when a single character searches a 10'x10' area of a dungeon he has only a 1/6 probability (excluding the possibility of demi-humans and thieves, which change the numbers, but not the essential idea) of finding a door that may or may not be there.  Given that, it seems insane how easily some players give up.  Even if there is a secret door, you're unlikely to find it given a single turn of searching by one character. 

There's two possible solutions.  Searching for multiple turns, and having multiple characters search the same area simultaneously.  The pay off is fairly quick.  Consider the case of multiple simultaneous searchers with no special abilities.




Figure 1:  Multiple simultaneous searchers with no special ability rapidly increase the likelihood of finding a secret door.

 Given the presense of a secret door, you're much more likely to find it now if everyone in a mid-sized party of 5 or 6 PCs searches the same area.  If a larger party of 9 or 10 people searches, you're really in business, finding more than 80% of any doors that exist in the room.  Elves, dwarves, thieves, magic and what not just make the picture even rosier.

The interesting question now, is how confident are you that there exists no secret door given that you haven't found a secret door.  That requires appealing to Bayes Theorem.  It also depends on the number of secret doors in the dungeon and the size of the dungeon.  I figured the Caves of Chaos was a fairly good dungeon to sample, so I just counted the number of squares in the goblin lair and found that only 2 out of 119 10'x10' squares had a secret door in them, so that means the probability of there being no secret door is >0.98.  Assuming that's an adequate "average" dungeon the following chart holds.

Figure 2:  With 3 or 4 characters you can be very sure that there's no secret door if you didn't find one.

The first interesting observation is that even with the dismal search performance of a single character, not finding a secret door still means it's overwhelmingly likely that there is in fact no secret door.  Ruling out doors isn't really a problem.  That's a function of the fact that most walls in a dungeon don't have secret doors.  The problem is that if there is one, you're not likely to find it either. That being said with 4 or 5 characters searching simultaneously in the same area you can almost certain there's no door while having better than 50/50 odds of finding those doors which are there.

A similar result can be achieved by spending time searching.  A single character searching 5 turns will have the same effect as 5 characters searching simultaneously.  The down side of this is that a dungeon master will most likely be rolling for wandering monsters during this time.  If this is the case, the other characters should stand guard.

I also wonder if this is a case where it's wise to designate a "caller" even though in most parties there isn't usually one regardless of the recommendation of the rule book.  Leadership and decision making pay off big time in the game.  Individualism tends to lead to lackluster performance.